Inverses of Exponential Functions - AP Precalculus
Card 1 of 30
Convert $y = 5^{x-4}$ to its inverse function.
Convert $y = 5^{x-4}$ to its inverse function.
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The inverse is $y = \text{log}_5(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.
The inverse is $y = \text{log}_5(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.
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What is the inverse of $f(x) = 8^{2x}$?
What is the inverse of $f(x) = 8^{2x}$?
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The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_8(x)$. Coefficient 2 in exponent becomes divisor $\frac{1}{2}$.
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_8(x)$. Coefficient 2 in exponent becomes divisor $\frac{1}{2}$.
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Find the inverse of $f(x) = 12^{2x+1}$.
Find the inverse of $f(x) = 12^{2x+1}$.
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The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}$. Factor out coefficient from exponent terms.
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}$. Factor out coefficient from exponent terms.
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Find the inverse of $f(x) = e^{x+2}$.
Find the inverse of $f(x) = e^{x+2}$.
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The inverse is $f^{-1}(x) = \text{ln}(x) - 2$. Addition in exponent becomes subtraction in inverse.
The inverse is $f^{-1}(x) = \text{ln}(x) - 2$. Addition in exponent becomes subtraction in inverse.
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Identify the inverse of $f(x) = b^{2x+1}$.
Identify the inverse of $f(x) = b^{2x+1}$.
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The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}$. Factor out coefficient and shift from exponent.
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}$. Factor out coefficient and shift from exponent.
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Identify the steps to find inverse of $f(x) = a^x$.
Identify the steps to find inverse of $f(x) = a^x$.
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Switch $x$ and $y$, solve for $y$. Result: $y = \text{log}_a(x)$. Standard method for finding inverse functions.
Switch $x$ and $y$, solve for $y$. Result: $y = \text{log}_a(x)$. Standard method for finding inverse functions.
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Which is the inverse of $f(x) = 9^{x-2}$?
Which is the inverse of $f(x) = 9^{x-2}$?
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The inverse is $f^{-1}(x) = \text{log}_9(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.
The inverse is $f^{-1}(x) = \text{log}_9(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.
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Find the inverse of $f(x) = 3^{x+5}$.
Find the inverse of $f(x) = 3^{x+5}$.
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The inverse is $f^{-1}(x) = \text{log}_3(x) - 5$. Addition in exponent becomes subtraction in inverse.
The inverse is $f^{-1}(x) = \text{log}_3(x) - 5$. Addition in exponent becomes subtraction in inverse.
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What is the inverse of $f(x) = 13^x$?
What is the inverse of $f(x) = 13^x$?
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The inverse is $f^{-1}(x) = \text{log}_{13}(x)$. Base-13 logarithm undoes base-13 exponentiation.
The inverse is $f^{-1}(x) = \text{log}_{13}(x)$. Base-13 logarithm undoes base-13 exponentiation.
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Identify the inverse of $f(x) = 9^{x-4}$.
Identify the inverse of $f(x) = 9^{x-4}$.
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The inverse is $f^{-1}(x) = \text{log}_9(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.
The inverse is $f^{-1}(x) = \text{log}_9(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.
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Convert $y = 2^{x-5}$ to its inverse function.
Convert $y = 2^{x-5}$ to its inverse function.
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The inverse is $y = \text{log}_2(x) + 5$. Subtract 5 from exponent, so add 5 to inverse.
The inverse is $y = \text{log}_2(x) + 5$. Subtract 5 from exponent, so add 5 to inverse.
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Find the inverse of $f(x) = 12^{2x+1}$.
Find the inverse of $f(x) = 12^{2x+1}$.
Tap to reveal answer
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}$. Factor out coefficient from exponent terms.
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}$. Factor out coefficient from exponent terms.
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Identify the inverse of $f(x) = 7^{3x-1}$.
Identify the inverse of $f(x) = 7^{3x-1}$.
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The inverse is $f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1)$. Factor out coefficient from both terms in exponent.
The inverse is $f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1)$. Factor out coefficient from both terms in exponent.
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Find the inverse of $f(x) = 11^{x+2}$.
Find the inverse of $f(x) = 11^{x+2}$.
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The inverse is $f^{-1}(x) = \text{log}_{11}(x) - 2$. Addition in exponent becomes subtraction in inverse.
The inverse is $f^{-1}(x) = \text{log}_{11}(x) - 2$. Addition in exponent becomes subtraction in inverse.
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Identify the inverse of $f(x) = b^{2x+1}$.
Identify the inverse of $f(x) = b^{2x+1}$.
Tap to reveal answer
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}$. Factor out coefficient and shift from exponent.
The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}$. Factor out coefficient and shift from exponent.
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Which function is the inverse of $f(x) = 4^{x-3}$?
Which function is the inverse of $f(x) = 4^{x-3}$?
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The inverse is $f^{-1}(x) = \text{log}_4(x) + 3$. Subtract 3 from exponent, so add 3 to inverse.
The inverse is $f^{-1}(x) = \text{log}_4(x) + 3$. Subtract 3 from exponent, so add 3 to inverse.
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Find the inverse of $f(x) = e^{x+2}$.
Find the inverse of $f(x) = e^{x+2}$.
Tap to reveal answer
The inverse is $f^{-1}(x) = \text{ln}(x) - 2$. Addition in exponent becomes subtraction in inverse.
The inverse is $f^{-1}(x) = \text{ln}(x) - 2$. Addition in exponent becomes subtraction in inverse.
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What is the inverse of $f(x) = 10^{3x}$?
What is the inverse of $f(x) = 10^{3x}$?
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The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.
The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.
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What is the inverse of $f(x) = 2^{3x}$?
What is the inverse of $f(x) = 2^{3x}$?
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The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_2(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.
The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_2(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.
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Identify the inverse of $f(x) = 15^{x-2}$.
Identify the inverse of $f(x) = 15^{x-2}$.
Tap to reveal answer
The inverse is $f^{-1}(x) = \text{log}_{15}(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.
The inverse is $f^{-1}(x) = \text{log}_{15}(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.
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What is the inverse of the function $f(x) = e^x$?
What is the inverse of the function $f(x) = e^x$?
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The inverse is $f^{-1}(x) = \text{ln}(x)$. Natural log undoes the natural exponential function.
The inverse is $f^{-1}(x) = \text{ln}(x)$. Natural log undoes the natural exponential function.
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What is the inverse of $f(x) = b^x$?
What is the inverse of $f(x) = b^x$?
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The inverse is $f^{-1}(x) = \text{log}_b(x)$. General base-$b$ logarithm undoes base-$b$ exponentiation.
The inverse is $f^{-1}(x) = \text{log}_b(x)$. General base-$b$ logarithm undoes base-$b$ exponentiation.
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Identify the inverse of $f(x) = 2^x$.
Identify the inverse of $f(x) = 2^x$.
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The inverse is $f^{-1}(x) = \text{log}_2(x)$. Base-2 logarithm undoes base-2 exponentiation.
The inverse is $f^{-1}(x) = \text{log}_2(x)$. Base-2 logarithm undoes base-2 exponentiation.
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State the property used to find inverses of exponential functions.
State the property used to find inverses of exponential functions.
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Exponential and logarithmic functions are inverses. Each function undoes the other's operation.
Exponential and logarithmic functions are inverses. Each function undoes the other's operation.
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What is the inverse of $f(x) = 10^x$?
What is the inverse of $f(x) = 10^x$?
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The inverse is $f^{-1}(x) = \text{log}_{10}(x)$. Common logarithm undoes base-10 exponentiation.
The inverse is $f^{-1}(x) = \text{log}_{10}(x)$. Common logarithm undoes base-10 exponentiation.
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Which function is the inverse of $f(x) = 3^x$?
Which function is the inverse of $f(x) = 3^x$?
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The inverse is $f^{-1}(x) = \text{log}_3(x)$. Base-3 logarithm undoes base-3 exponentiation.
The inverse is $f^{-1}(x) = \text{log}_3(x)$. Base-3 logarithm undoes base-3 exponentiation.
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What operation is the inverse of exponentiation?
What operation is the inverse of exponentiation?
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The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.
The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.
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Find the inverse of $f(x) = 5^x$.
Find the inverse of $f(x) = 5^x$.
Tap to reveal answer
The inverse is $f^{-1}(x) = \text{log}_5(x)$. Base-5 logarithm undoes base-5 exponentiation.
The inverse is $f^{-1}(x) = \text{log}_5(x)$. Base-5 logarithm undoes base-5 exponentiation.
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What is the inverse of the natural exponential function?
What is the inverse of the natural exponential function?
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The inverse is the natural logarithm function. Natural log $\ln(x)$ undoes $e^x$.
The inverse is the natural logarithm function. Natural log $\ln(x)$ undoes $e^x$.
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Identify the inverse of $f(x) = a^x$.
Identify the inverse of $f(x) = a^x$.
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The inverse is $f^{-1}(x) = \text{log}_a(x)$. Base-$a$ logarithm undoes base-$a$ exponentiation.
The inverse is $f^{-1}(x) = \text{log}_a(x)$. Base-$a$ logarithm undoes base-$a$ exponentiation.
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