Algebra 2 Flashcards: Solving Exponential Equations With Logarithms

What this deck covers

This deck focuses on Solving Exponential Equations With Logarithms, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Rewrite the solution of $a\cdot 10^{ct}=d$ using $\ln$ only.

Answer: $t=\frac{\ln\left(\frac{d}{a}\right)}{c\ln(10)}$. Apply change-of-base formula: $\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$.

Flashcard 2: State the change-of-base formula for rewriting $\log_b(x)$ using $\log$.

Answer: $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any logarithm base to common logarithm form.

Flashcard 3: Find $t$ in $10^{3t}=10^{12}$ and give the exact value.

Answer: $t=4$. Since the bases are equal, equate exponents: $3t = 12$, so $t = 4$.

Flashcard 4: Find $t$ in $4\cdot 2^{t}=64$ and express the result as a logarithm.

Answer: $t=\log_2(16)$. Divide by 4: $2^t = 16$, then take base-2 logarithm.

Flashcard 5: Find $t$ in $0.2\cdot 10^{4t}=2$ and express the result as a logarithm.

Answer: $t=\frac{\log(10)}{4}$. Multiply by 5: $10^{4t} = 10$, then take log and divide by 4.

Flashcard 6: Find $t$ in $5\cdot 10^{2t}=500$ and give the exact value.

Answer: $t=1$. Since $\log(100) = 2$ and we divide by 2, we get $t = 1$.

Flashcard 7: Identify the final step to isolate $t$ from $ct=\log_b\left(\frac{d}{a}\right)$.

Answer: Divide by $c$: $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Divide both sides by coefficient $c$ to solve for $t$.

Flashcard 8: Find $t$ in $2\cdot e^{t}=10$ and express the result as a logarithm.

Answer: $t=\ln(5)$. Divide by 2: $e^t = 5$, then take natural logarithm.

Flashcard 9: What is the solution for $t$ in $a\cdot 10^{ct}=d$ written using common logarithm?

Answer: $t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $10^{ct} = \frac{d}{a}$, then apply common log.

Flashcard 10: Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.

Answer: $t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.

Flashcard 11: Identify the correct log form of the solution to $a\cdot e^{ct}=d$.

Answer: $t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Standard form using natural logarithm for base $e$ exponentials.

Flashcard 12: Rewrite the solution of $a\cdot 2^{ct}=d$ using $\ln$ only.

Answer: $t=\frac{\ln\left(\frac{d}{a}\right)}{c\ln(2)}$. Apply change-of-base formula: $\log_2(x) = \frac{\ln(x)}{\ln(2)}$.

Flashcard 13: Find $t$ in $0.1\cdot 10^{t}=1$ and express the result as a logarithm.

Answer: $t=\log(10)$. Multiply by 10: $10^t = 10$, so $t = \log(10)$.

Flashcard 14: Find $t$ in $0.5\cdot e^{t}=4$ and express the result as a logarithm.

Answer: $t=\ln(8)$. Multiply by 2: $e^t = 8$, then take natural logarithm.

Flashcard 15: What property lets you move an exponent out front: $\log_b(x^k)=?$

Answer: $\log_b(x^k)=k\log_b(x)$. Uses the power property to bring exponents in front.

Flashcard 16: Find $t$ in $1.5\cdot 2^{0.5t}=6$ and express the result as a logarithm.

Answer: $t=\frac{\log_2(4)}{0.5}$. Divide by 1.5: $2^{0.5t} = 4$, then take log and divide by 0.5.

Flashcard 17: Find $t$ in $6\cdot 2^{t}=3$ and express the result as a logarithm.

Answer: $t=\log_2\left(\frac{1}{2}\right)$. Divide by 6: $2^t = \frac{1}{2}$, then take base-2 logarithm.

Flashcard 18: Find $t$ in $3\cdot 2^{t}=24$ and give the exact value.

Answer: $t=3$. Since $\log_2(8) = 3$, we get $t = 3$.

Flashcard 19: Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?

Answer: $y=\log_b(x)$. Logarithm is the inverse function of exponentiation.

Flashcard 20: What is the solution for $t$ in $a\cdot e^{ct}=d$ written using natural logarithm?

Answer: $t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $e^{ct} = \frac{d}{a}$, then apply natural log.

Flashcard 21: Find $t$ in $0.01\cdot 10^{t}=100$ and express the result as a logarithm.

Answer: $t=\log(10000)$. Multiply by 100: $10^t = 10000$, so $t = \log(10000)$.

Flashcard 22: Identify the correct log form of the solution to $a\cdot 2^{ct}=d$.

Answer: $t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Standard form using base-2 logarithm for base 2 exponentials.

Flashcard 23: Find $t$ in $5\cdot e^{2t}=40$ and express the result as a logarithm.

Answer: $t=\frac{\ln(8)}{2}$. Divide by 5: $e^{2t} = 8$, then take ln and divide by 2.

Flashcard 24: Find $t$ in $3\cdot 2^{2t}=96$ and express the result as a logarithm.

Answer: $t=\frac{\log_2(32)}{2}$. Divide by 3: $2^{2t} = 32$, then take log and divide by 2.

Flashcard 25: State the change-of-base formula for rewriting $\log_b(x)$ using $\ln$.

Answer: $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any logarithm base to natural logarithm form.

Flashcard 26: Find $t$ in $e^{2t}=e^{6}$ and give the exact value.

Answer: $t=3$. Since the bases are equal, equate exponents: $2t = 6$, so $t = 3$.

Flashcard 27: Find $t$ in $7\cdot e^{3t}=1$ and express the result as a logarithm.

Answer: $t=\frac{\ln\left(\frac{1}{7}\right)}{3}$. Divide by 7: $e^{3t} = \frac{1}{7}$, then take ln and divide by 3.

Flashcard 28: Find $t$ in $9\cdot 2^{3t}=72$ and express the result as a logarithm.

Answer: $t=\frac{\log_2(8)}{3}$. Divide by 9: $2^{3t} = 8$, then take log and divide by 3.

Flashcard 29: What base restriction is required for $\log_b(x)$ when solving $a b^{ct}=d$?

Answer: $b>0$ and $b\ne 1$. Base must be positive and not equal to 1 for valid logarithm.

Flashcard 30: Find $t$ in $3\cdot e^{0.1t}=9$ and express the result as a logarithm.

Answer: $t=\frac{\ln(3)}{0.1}$. Divide by 3: $e^{0.1t} = 3$, then take ln and divide by 0.1.

Flashcard 31: Find $t$ in $8\cdot 10^{-t}=2$ and express the result as a logarithm.

Answer: $t=\frac{\log\left(\frac{1}{4}\right)}{-1}$. Divide by 8: $10^{-t} = \frac{1}{4}$, then take log and divide by -1.

Flashcard 32: What is the solution for $t$ in $a\cdot 2^{ct}=d$ written using base-$2$ logarithm?

Answer: $t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $2^{ct} = \frac{d}{a}$, then apply base-2 log.

Flashcard 33: Find $t$ in $3\cdot 10^{2t}=300$ and express the result as a logarithm.

Answer: $t=\frac{\log(100)}{2}$. Divide by 3: $10^{2t} = 100$, then take log and divide by 2.

Flashcard 34: Find $t$ in $7\cdot 10^{t}=1.4$ and express the result as a logarithm.

Answer: $t=\log\left(\frac{1}{5}\right)$. Divide by 7: $10^t = \frac{1.4}{7} = 0.2 = \frac{1}{5}$, then take log.

Flashcard 35: Find $t$ in $7\cdot e^{t}=7$ and give the exact value.

Answer: $t=0$. Since $\ln(1) = 0$ when $e^t = 1$, we get $t = 0$.

Flashcard 36: Rewrite the solution of $a\cdot e^{ct}=d$ using $\log$ only.

Answer: $t=\frac{\log\left(\frac{d}{a}\right)}{c\log(e)}$. Apply change-of-base formula: $\ln(x) = \frac{\log(x)}{\log(e)}$.

Flashcard 37: Find $t$ in $4\cdot 10^{t}=40$ and give the exact value.

Answer: $t=1$. Since $\log(10) = 1$, we get $t = 1$.

Flashcard 38: Find $t$ in $2\cdot 2^{t}=64$ and give the exact value.

Answer: $t=5$. Since $\log_2(32) = 5$, we get $t = 5$.

Flashcard 39: Identify the correct log form of the solution to $a\cdot 10^{ct}=d$.

Answer: $t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Standard form using common logarithm for base 10 exponentials.

Flashcard 40: Find $t$ in $5\cdot 10^{t}=200$ and express the result as a logarithm.

Answer: $t=\log(40)$. Divide by 5: $10^t = 40$, then take common logarithm.

Flashcard 41: What property lets you rewrite $\log_b\left(\frac{d}{a}\right)$ as a difference of logs?

Answer: $\log_b\left(\frac{d}{a}\right)=\log_b(d)-\log_b(a)$. Uses the quotient property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.

Flashcard 42: What is the domain requirement for $\log_b(x)$ in solving exponential equations?

Answer: $x>0$. Logarithm is only defined for positive arguments.

Flashcard 43: Find $t$ in $2^{4t}=2^{20}$ and give the exact value.

Answer: $t=5$. Since the bases are equal, equate exponents: $4t = 20$, so $t = 5$.

Flashcard 44: Find $t$ in $0.25\cdot 2^{2t}=8$ and express the result as a logarithm.

Answer: $t=\frac{\log_2(32)}{2}$. Multiply by 4: $2^{2t} = 32$, then take log and divide by 2.

Flashcard 45: Find $t$ in $0.5\cdot 2^{t}=1$ and express the result as a logarithm.

Answer: $t=\log_2(2)$. Multiply by 2: $2^t = 2$, so $t = \log_2(2)$.

Flashcard 46: Find $t$ in $2\cdot e^{-t}=8$ and express the result as a logarithm.

Answer: $t=\frac{\ln(4)}{-1}$. Divide by 2: $e^{-t} = 4$, then take ln and divide by -1.

Flashcard 47: Find $t$ in $10\cdot e^{-0.2t}=5$ and express the result as a logarithm.

Answer: $t=\frac{\ln\left(\frac{1}{2}\right)}{-0.2}$. Divide by 10: $e^{-0.2t} = \frac{1}{2}$, then take ln and divide by -0.2.

Flashcard 48: What log step isolates $ct$ after dividing: $b^{ct}=\frac{d}{a}$?

Answer: $ct=\log_b\left(\frac{d}{a}\right)$. Take $\log_b$ of both sides to eliminate the exponential.

Flashcard 49: Find $t$ in $9\cdot 2^{-t}=72$ and express the result as a logarithm.

Answer: $t=\frac{\log_2(8)}{-1}$. Divide by 9: $2^{-t} = 8$, then take log and divide by -1.

Flashcard 50: Find $t$ in $16\cdot 2^{t}=1$ and express the result as a logarithm.

Answer: $t=\log_2\left(\frac{1}{16}\right)$. Divide by 16: $2^t = \frac{1}{16}$, then take base-2 log.

Flashcard 51: Identify the first algebra step to solve $a b^{ct}=d$ for $t$.

Answer: Divide: $b^{ct}=\frac{d}{a}$. Isolate the exponential term before taking logarithms.

Flashcard 52: Find $t$ in $12\cdot 10^{0.5t}=3$ and express the result as a logarithm.

Answer: $t=\frac{\log\left(\frac{1}{4}\right)}{0.5}$. Divide by 12: $10^{0.5t} = \frac{1}{4}$, then take log and divide by 0.5.

Flashcard 53: State the solution for $t$ in $a b^{ct}=d$ written using a logarithm.

Answer: $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Isolate $b^{ct}$ by dividing by $a$, then take $\log_b$ of both sides.