Solving Exponential Equations with Logarithms - Algebra 2
Card 1 of 30
Identify the correct log form of the solution to $a\cdot 10^{ct}=d$.
Identify the correct log form of the solution to $a\cdot 10^{ct}=d$.
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$t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Standard form using common logarithm for base 10 exponentials.
$t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Standard form using common logarithm for base 10 exponentials.
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Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?
Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?
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$y=\log_b(x)$. Logarithm is the inverse function of exponentiation.
$y=\log_b(x)$. Logarithm is the inverse function of exponentiation.
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Find $t$ in $5\cdot 10^{t}=200$ and express the result as a logarithm.
Find $t$ in $5\cdot 10^{t}=200$ and express the result as a logarithm.
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$t=\log(40)$. Divide by 5: $10^t = 40$, then take common logarithm.
$t=\log(40)$. Divide by 5: $10^t = 40$, then take common logarithm.
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Find $t$ in $4\cdot 10^{t}=40$ and give the exact value.
Find $t$ in $4\cdot 10^{t}=40$ and give the exact value.
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$t=1$. Since $\log(10) = 1$, we get $t = 1$.
$t=1$. Since $\log(10) = 1$, we get $t = 1$.
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Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.
Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.
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$t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.
$t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.
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Identify the final step to isolate $t$ from $ct=\log_b\left(\frac{d}{a}\right)$.
Identify the final step to isolate $t$ from $ct=\log_b\left(\frac{d}{a}\right)$.
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Divide by $c$: $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Divide both sides by coefficient $c$ to solve for $t$.
Divide by $c$: $t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Divide both sides by coefficient $c$ to solve for $t$.
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Find $t$ in $2\cdot e^{-t}=8$ and express the result as a logarithm.
Find $t$ in $2\cdot e^{-t}=8$ and express the result as a logarithm.
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$t=\frac{\ln(4)}{-1}$. Divide by 2: $e^{-t} = 4$, then take ln and divide by -1.
$t=\frac{\ln(4)}{-1}$. Divide by 2: $e^{-t} = 4$, then take ln and divide by -1.
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State the solution for $t$ in $a b^{ct}=d$ written using a logarithm.
State the solution for $t$ in $a b^{ct}=d$ written using a logarithm.
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$t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Isolate $b^{ct}$ by dividing by $a$, then take $\log_b$ of both sides.
$t=\frac{\log_b\left(\frac{d}{a}\right)}{c}$. Isolate $b^{ct}$ by dividing by $a$, then take $\log_b$ of both sides.
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What is the solution for $t$ in $a\cdot 10^{ct}=d$ written using common logarithm?
What is the solution for $t$ in $a\cdot 10^{ct}=d$ written using common logarithm?
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$t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $10^{ct} = \frac{d}{a}$, then apply common log.
$t=\frac{\log\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $10^{ct} = \frac{d}{a}$, then apply common log.
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What is the solution for $t$ in $a\cdot e^{ct}=d$ written using natural logarithm?
What is the solution for $t$ in $a\cdot e^{ct}=d$ written using natural logarithm?
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$t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $e^{ct} = \frac{d}{a}$, then apply natural log.
$t=\frac{\ln\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $e^{ct} = \frac{d}{a}$, then apply natural log.
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What is the solution for $t$ in $a\cdot 2^{ct}=d$ written using base-$2$ logarithm?
What is the solution for $t$ in $a\cdot 2^{ct}=d$ written using base-$2$ logarithm?
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$t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $2^{ct} = \frac{d}{a}$, then apply base-2 log.
$t=\frac{\log_2\left(\frac{d}{a}\right)}{c}$. Divide by $a$ to get $2^{ct} = \frac{d}{a}$, then apply base-2 log.
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State the change-of-base formula for rewriting $\log_b(x)$ using $\ln$.
State the change-of-base formula for rewriting $\log_b(x)$ using $\ln$.
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$\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any logarithm base to natural logarithm form.
$\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any logarithm base to natural logarithm form.
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State the change-of-base formula for rewriting $\log_b(x)$ using $\log$.
State the change-of-base formula for rewriting $\log_b(x)$ using $\log$.
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$\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any logarithm base to common logarithm form.
$\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any logarithm base to common logarithm form.
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What property lets you rewrite $\log_b\left(\frac{d}{a}\right)$ as a difference of logs?
What property lets you rewrite $\log_b\left(\frac{d}{a}\right)$ as a difference of logs?
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$\log_b\left(\frac{d}{a}\right)=\log_b(d)-\log_b(a)$. Uses the quotient property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.
$\log_b\left(\frac{d}{a}\right)=\log_b(d)-\log_b(a)$. Uses the quotient property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.
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What property lets you move an exponent out front: $\log_b(x^k)=?$
What property lets you move an exponent out front: $\log_b(x^k)=?$
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$\log_b(x^k)=k\log_b(x)$. Uses the power property to bring exponents in front.
$\log_b(x^k)=k\log_b(x)$. Uses the power property to bring exponents in front.
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Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?
Identify the inverse statement that justifies taking logs: $b^{y}=x$ implies what?
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$y=\log_b(x)$. Logarithm is the inverse function of exponentiation.
$y=\log_b(x)$. Logarithm is the inverse function of exponentiation.
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What is the domain requirement for $\log_b(x)$ in solving exponential equations?
What is the domain requirement for $\log_b(x)$ in solving exponential equations?
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$x>0$. Logarithm is only defined for positive arguments.
$x>0$. Logarithm is only defined for positive arguments.
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What base restriction is required for $\log_b(x)$ when solving $a b^{ct}=d$?
What base restriction is required for $\log_b(x)$ when solving $a b^{ct}=d$?
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$b>0$ and $b\ne 1$. Base must be positive and not equal to 1 for valid logarithm.
$b>0$ and $b\ne 1$. Base must be positive and not equal to 1 for valid logarithm.
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Find $t$ in $3\cdot 10^{2t}=300$ and express the result as a logarithm.
Find $t$ in $3\cdot 10^{2t}=300$ and express the result as a logarithm.
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$t=\frac{\log(100)}{2}$. Divide by 3: $10^{2t} = 100$, then take log and divide by 2.
$t=\frac{\log(100)}{2}$. Divide by 3: $10^{2t} = 100$, then take log and divide by 2.
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Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.
Find $t$ in $2\cdot 10^{3t}=50$ and express the result as a logarithm.
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$t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.
$t=\frac{\log(25)}{3}$. Divide by 2: $10^{3t} = 25$, then take log and divide by 3.
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Find $t$ in $7\cdot 10^{t}=1.4$ and express the result as a logarithm.
Find $t$ in $7\cdot 10^{t}=1.4$ and express the result as a logarithm.
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$t=\log\left(\frac{1}{5}\right)$. Divide by 7: $10^t = \frac{1.4}{7} = 0.2 = \frac{1}{5}$, then take log.
$t=\log\left(\frac{1}{5}\right)$. Divide by 7: $10^t = \frac{1.4}{7} = 0.2 = \frac{1}{5}$, then take log.
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Find $t$ in $4\cdot 2^{t}=64$ and express the result as a logarithm.
Find $t$ in $4\cdot 2^{t}=64$ and express the result as a logarithm.
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$t=\log_2(16)$. Divide by 4: $2^t = 16$, then take base-2 logarithm.
$t=\log_2(16)$. Divide by 4: $2^t = 16$, then take base-2 logarithm.
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Find $t$ in $3\cdot 2^{2t}=96$ and express the result as a logarithm.
Find $t$ in $3\cdot 2^{2t}=96$ and express the result as a logarithm.
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$t=\frac{\log_2(32)}{2}$. Divide by 3: $2^{2t} = 32$, then take log and divide by 2.
$t=\frac{\log_2(32)}{2}$. Divide by 3: $2^{2t} = 32$, then take log and divide by 2.
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Find $t$ in $6\cdot 2^{t}=3$ and express the result as a logarithm.
Find $t$ in $6\cdot 2^{t}=3$ and express the result as a logarithm.
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$t=\log_2\left(\frac{1}{2}\right)$. Divide by 6: $2^t = \frac{1}{2}$, then take base-2 logarithm.
$t=\log_2\left(\frac{1}{2}\right)$. Divide by 6: $2^t = \frac{1}{2}$, then take base-2 logarithm.
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Find $t$ in $9\cdot 2^{3t}=72$ and express the result as a logarithm.
Find $t$ in $9\cdot 2^{3t}=72$ and express the result as a logarithm.
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$t=\frac{\log_2(8)}{3}$. Divide by 9: $2^{3t} = 8$, then take log and divide by 3.
$t=\frac{\log_2(8)}{3}$. Divide by 9: $2^{3t} = 8$, then take log and divide by 3.
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Find $t$ in $2\cdot e^{t}=10$ and express the result as a logarithm.
Find $t$ in $2\cdot e^{t}=10$ and express the result as a logarithm.
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$t=\ln(5)$. Divide by 2: $e^t = 5$, then take natural logarithm.
$t=\ln(5)$. Divide by 2: $e^t = 5$, then take natural logarithm.
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Find $t$ in $5\cdot e^{2t}=40$ and express the result as a logarithm.
Find $t$ in $5\cdot e^{2t}=40$ and express the result as a logarithm.
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$t=\frac{\ln(8)}{2}$. Divide by 5: $e^{2t} = 8$, then take ln and divide by 2.
$t=\frac{\ln(8)}{2}$. Divide by 5: $e^{2t} = 8$, then take ln and divide by 2.
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Find $t$ in $7\cdot e^{3t}=1$ and express the result as a logarithm.
Find $t$ in $7\cdot e^{3t}=1$ and express the result as a logarithm.
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$t=\frac{\ln\left(\frac{1}{7}\right)}{3}$. Divide by 7: $e^{3t} = \frac{1}{7}$, then take ln and divide by 3.
$t=\frac{\ln\left(\frac{1}{7}\right)}{3}$. Divide by 7: $e^{3t} = \frac{1}{7}$, then take ln and divide by 3.
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Find $t$ in $0.5\cdot e^{t}=4$ and express the result as a logarithm.
Find $t$ in $0.5\cdot e^{t}=4$ and express the result as a logarithm.
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$t=\ln(8)$. Multiply by 2: $e^t = 8$, then take natural logarithm.
$t=\ln(8)$. Multiply by 2: $e^t = 8$, then take natural logarithm.
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Find $t$ in $12\cdot 10^{0.5t}=3$ and express the result as a logarithm.
Find $t$ in $12\cdot 10^{0.5t}=3$ and express the result as a logarithm.
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$t=\frac{\log\left(\frac{1}{4}\right)}{0.5}$. Divide by 12: $10^{0.5t} = \frac{1}{4}$, then take log and divide by 0.5.
$t=\frac{\log\left(\frac{1}{4}\right)}{0.5}$. Divide by 12: $10^{0.5t} = \frac{1}{4}$, then take log and divide by 0.5.
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