This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $2·e^{0.5t}$=20, divide by 2: $e^{0.5t}$=10, take natural log: $ln(e^{0.5t}$)=ln(10), simplify to 0.5t=ln(10), and multiply by 2: t=2 ln(10). Choice A correctly isolates, takes the natural logarithm, applies the inverse property since $ln(e^y$)=y, and approximates to 4.605. Choice B forgets to multiply by 2, resulting in t=ln(10) which is half the correct value—be sure to solve for t by dividing by the coefficient 0.5, which is the same as multiplying by 2! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $4·2^{t/5}$=30, divide by 4: $2^{t/5}$=30/4=15/2, take log base 2: $log₂(2^{t/5}$)=log₂(15/2), simplify to t/5=log₂(15/2), and multiply by 5: t=5 log₂(15/2). Choice A correctly isolates, takes log base 2, applies the inverse property, and approximates to 14.535 using change of base on a calculator. Choice B divides by 5 instead of multiplying, resulting in t=log₂(15/2)/5 which is one-fifth the size—remember that to undo division in the exponent, you multiply when solving for t! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $8·2^{x/2}$=70, divide by 8: $2^{x/2}$=70/8=35/4, take log base 2: $log₂(2^{x/2}$)=log₂(35/4), simplify to x/2=log₂(35/4), and multiply by 2: x=2 log₂(35/4). Choice A correctly isolates the exponential, takes log base 2, applies the inverse property, and approximates to 6.258 using change of base. Choice B divides by 2 instead of multiplying, giving x=log₂(35/4)/2 which is half the correct value—remember to multiply by 2 to undo the division in the exponent! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $3·10^{2x}$=75, first divide both sides by 3 to isolate the exponential: $10^{2x}$=25, then take the common logarithm (base 10) of both sides: $log(10^{2x}$)=log(25), which simplifies to 2x=log(25) using the inverse property, and finally divide by 2: x=log(25)/2. Choice A correctly isolates and takes the logarithm to get the exact solution as log(25)/2 and approximates it to 0.699 using technology. Choice C forgets to divide by 2 after taking the log, resulting in x=log(25) which is twice as large as it should be—remember to solve fully for the variable after applying the logarithm! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $7·10^{t/3}$=50, divide by 7: $10^{t/3}$=50/7, take common log: $log(10^{t/3}$)=log(50/7), simplify to t/3=log(50/7), and multiply by 3: t=3 log(50/7). Choice A correctly isolates the exponential, takes the common logarithm, uses the inverse property, and provides the exact form with an approximation. Choice B skips multiplying by 3, giving t=log(50/7) which is one-third the correct value—don't forget to multiply by the reciprocal of the exponent's coefficient! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $2.5·e^{t/4}$=40, divide by 2.5: $e^{t/4}$=16, take natural log: $ln(e^{t/4}$)=ln(16), simplify to t/4=ln(16), and multiply by 4: t=4 ln(16). Choice A correctly isolates, takes the natural logarithm, uses the inverse property, and approximates to 11.090 using technology. Choice B divides by 4 instead of multiplying, resulting in t=ln(16)/4 which is one-fourth the value—always multiply by the reciprocal of the exponent's coefficient! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for $ab^{ct} = d$: (1) Isolate the exponential by dividing both sides by $a$: $b^{ct} = \frac{d}{a}$, (2) Take log base $b$ of both sides: $\log_b(b^{ct}) = \log_b(\frac{d}{a})$, (3) Use inverse property on left: $ct = \log_b(\frac{d}{a})$, (4) Solve for variable: $t = \frac{\log_b(\frac{d}{a})}{c}$. This systematic approach works for any exponential equation in this form! To solve $12 \cdot e^{2x} = 90$, divide by 12: $e^{2x} = \frac{90}{12} = \frac{15}{2}$, take natural log: $\ln(e^{2x}) = \ln(\frac{15}{2})$, simplify to $2x = \ln(\frac{15}{2})$, and divide by 2: $x = \frac{\ln(\frac{15}{2})}{2}$. Choice A correctly isolates the exponential, takes the natural logarithm, applies the inverse property, and approximates to 1.007. Choice C omits dividing by 2, giving $x = \ln(\frac{15}{2})$ which is twice the correct value—always divide by the exponent's coefficient to isolate the variable! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need $\log_2(10)$? Use change of base: $\log_2(10) = \frac{\ln(10)}{\ln(2)}$ or $\frac{\log(10)}{\log(2)}$—both give the same answer ≈ 3.322. The formula is $\log_b(x) = \frac{\ln(x)}{\ln(b)}$ for any base $b$. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression $b^{ct}$ before taking logarithms. If you have $5 \cdot 2^t = 40$, first divide by 5 to get $2^t = 8$, THEN take log. Taking $\log_2$ of both sides of $5 \cdot 2^t = 40$ directly leads to $\log_2(5 \cdot 2^t)$, which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have $b^{\text{something}} = \text{number}$, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $5·2^{3t}$=60, first divide both sides by 5 to isolate: $2^{3t}$=12, then take log base 2: $log₂(2^{3t}$)=log₂(12), which simplifies to 3t=log₂(12), and divide by 3: t=log₂(12)/3. Choice A correctly isolates the exponential, applies the logarithm base 2, uses the inverse property, and approximates to 1.195. Choice C omits dividing by 3, leaving t=log₂(12) which is three times larger than needed—always remember to divide by the coefficient in the exponent after taking the log! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. When solving exponential equations like $2^t$ = 10 where the exponent contains the variable, logarithms are the tool that unlocks the solution: taking log base 2 of both sides gives $log₂(2^t$) = log₂(10), and using the inverse property $log₂(2^t$) = t, we get t = log₂(10). This is the exact solution! To get a decimal approximation, use your calculator with change of base: log₂(10) = ln(10)/ln(2) ≈ 3.322. To solve $9·2^{4t}$=100, divide by 9: $2^{4t}$=100/9, take log base 2: $log₂(2^{4t}$)=log₂(100/9), simplify to 4t=log₂(100/9), and divide by 4: t=log₂(100/9)/4. Choice A correctly isolates, takes log base 2, uses the inverse property, and approximates to 0.869 using change of base. Choice D forgets to divide by 4, resulting in t=log₂(100/9) which is four times larger—make sure to divide by the coefficient in the exponent! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!

This question tests your ability to solve exponential equations by taking logarithms of both sides and using the inverse relationship to isolate the variable. The solving strategy for ab^(ct) = d: (1) Isolate the exponential by dividing both sides by a: b^(ct) = d/a, (2) Take log base b of both sides: log_b(b^(ct)) = log_b(d/a), (3) Use inverse property on left: ct = log_b(d/a), (4) Solve for variable: t = log_b(d/a)/c. This systematic approach works for any exponential equation in this form! To solve $6·10^{0.2t}$=48, divide by 6: $10^{0.2t}$=8, take common log: $log(10^{0.2t}$)=log(8), simplify to 0.2t=log(8), and divide by 0.2 (or multiply by 5): t=5 log(8). Choice A correctly isolates the exponential, takes the common logarithm, applies the inverse property, and approximates to 4.515. Choice B divides by 5 instead of multiplying, giving t=log(8)/5 which is too small—remember that dividing by 0.2 is multiplying by 5 to solve for t! Calculator strategy for non-standard bases: your calculator has log (base 10) and ln (base e) buttons, but what if you need log₂(10)? Use change of base: log₂(10) = ln(10)/ln(2) or log(10)/log(2)—both give the same answer ≈ 3.322. The formula is log_b(x) = ln(x)/ln(b) for any base b. This lets you evaluate any logarithm using just the ln button! Alternatively, leave answers in exact log form if calculator evaluation isn't required. Isolation before logarithms: ALWAYS isolate the exponential expression b^(ct) before taking logarithms. If you have $5·2^t$ = 40, first divide by 5 to get $2^t$ = 8, THEN take log. Taking log₂ of both sides of $5·2^t$ = 40 directly leads to $log₂(5·2^t$), which is more complex (requires log properties). Simple isolation first makes the logarithm application clean: take log of both sides when you have b^(something) = number, with the exponential alone on one side!