Exponential and Logarithmic Equations and Inequalities - AP Precalculus
Card 1 of 30
Solve the inequality: $\text{ln}(x) < 0$.
Solve the inequality: $\text{ln}(x) < 0$.
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$0 < x < 1$. Natural log is negative when input is between 0 and 1.
$0 < x < 1$. Natural log is negative when input is between 0 and 1.
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Simplify: $e^{\text{ln}(x)}$.
Simplify: $e^{\text{ln}(x)}$.
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$x$. Exponential and natural logarithm cancel as inverse functions.
$x$. Exponential and natural logarithm cancel as inverse functions.
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What is the base of the common logarithm?
What is the base of the common logarithm?
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- Common logarithm uses base 10 by convention.
- Common logarithm uses base 10 by convention.
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Solve for $x$: $\text{log}_2(x) = 3$.
Solve for $x$: $\text{log}_2(x) = 3$.
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$x = 8$. Convert to exponential form: $2^3 = 8$, so $x = 8$.
$x = 8$. Convert to exponential form: $2^3 = 8$, so $x = 8$.
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Find the solution: $2^x = 16$.
Find the solution: $2^x = 16$.
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$x = 4$. Since $16 = 2^4$, we have $x = 4$.
$x = 4$. Since $16 = 2^4$, we have $x = 4$.
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Identify the change of base formula for logarithms.
Identify the change of base formula for logarithms.
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$\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts logarithm from base $b$ to any other base $k$.
$\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts logarithm from base $b$ to any other base $k$.
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What is the general form of an exponential function?
What is the general form of an exponential function?
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$f(x) = ab^x$ where $a \neq 0$, $b > 0$, $b \neq 1$. Standard form where $a$ is initial value and $b$ is growth/decay factor.
$f(x) = ab^x$ where $a \neq 0$, $b > 0$, $b \neq 1$. Standard form where $a$ is initial value and $b$ is growth/decay factor.
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State the formula for the natural exponential function.
State the formula for the natural exponential function.
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$f(x) = e^x$. Uses Euler's number $e$ as the base for the exponential function.
$f(x) = e^x$. Uses Euler's number $e$ as the base for the exponential function.
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Define the natural logarithm function.
Define the natural logarithm function.
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$\text{ln}(x) = \text{log}_e(x)$. Natural log uses base $e$ (Euler's number) as the base.
$\text{ln}(x) = \text{log}_e(x)$. Natural log uses base $e$ (Euler's number) as the base.
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What is the range of $f(x) = b^x$ for $b > 1$?
What is the range of $f(x) = b^x$ for $b > 1$?
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$(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.
$(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.
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Solve for $x$: $5^x = 25$.
Solve for $x$: $5^x = 25$.
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$x = 2$. Since $25 = 5^2$, we have $x = 2$.
$x = 2$. Since $25 = 5^2$, we have $x = 2$.
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State the base of the natural logarithm.
State the base of the natural logarithm.
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$e$. Natural logarithm uses Euler's number $e ≈ 2.718$ as base.
$e$. Natural logarithm uses Euler's number $e ≈ 2.718$ as base.
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Solve for $x$: $e^x = 1$.
Solve for $x$: $e^x = 1$.
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$x = 0$. Any number to the power of 0 equals 1.
$x = 0$. Any number to the power of 0 equals 1.
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Simplify: $\text{log}_b(b^x)$.
Simplify: $\text{log}_b(b^x)$.
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$x$. Logarithm and exponential cancel each other as inverse functions.
$x$. Logarithm and exponential cancel each other as inverse functions.
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Convert to exponential form: $\text{log}_b(x) = y$.
Convert to exponential form: $\text{log}_b(x) = y$.
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$b^y = x$. Definition of logarithm as the inverse of exponential function.
$b^y = x$. Definition of logarithm as the inverse of exponential function.
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What is the inverse of $\text{ln}(x)$?
What is the inverse of $\text{ln}(x)$?
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$e^x$. Natural logarithm and exponential function are inverses.
$e^x$. Natural logarithm and exponential function are inverses.
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Solve the equation: $\text{ln}(x) = 2$.
Solve the equation: $\text{ln}(x) = 2$.
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$x = e^2$. Convert to exponential form: $e^2 = x$.
$x = e^2$. Convert to exponential form: $e^2 = x$.
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What is the value of $\text{log}_b(b^0)$?
What is the value of $\text{log}_b(b^0)$?
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- Since $b^0 = 1$ and $\log_b(1) = 0$.
- Since $b^0 = 1$ and $\log_b(1) = 0$.
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Identify the property: $b^{\text{log}_b(x)}$.
Identify the property: $b^{\text{log}_b(x)}$.
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$x$. Identity property: base raised to its own logarithm.
$x$. Identity property: base raised to its own logarithm.
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What is the domain of $\text{log}_b(x)$?
What is the domain of $\text{log}_b(x)$?
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$x > 0$. Logarithm only defined for positive real numbers.
$x > 0$. Logarithm only defined for positive real numbers.
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Solve the inequality: $3^x > 9$.
Solve the inequality: $3^x > 9$.
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$x > 2$. Since $9 = 3^2$, we need $x > 2$.
$x > 2$. Since $9 = 3^2$, we need $x > 2$.
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Solve for $x$: $4^x = 64$.
Solve for $x$: $4^x = 64$.
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$x = 3$. Since $64 = 4^3$, we have $x = 3$.
$x = 3$. Since $64 = 4^3$, we have $x = 3$.
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What is the base change formula for $\text{log}_b(x)$?
What is the base change formula for $\text{log}_b(x)$?
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$\text{log}_b(x) = \frac{\text{ln}(x)}{\text{ln}(b)}$. Uses natural logarithm to convert between bases.
$\text{log}_b(x) = \frac{\text{ln}(x)}{\text{ln}(b)}$. Uses natural logarithm to convert between bases.
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What is the range of the function $f(x) = \text{log}_b(x)$?
What is the range of the function $f(x) = \text{log}_b(x)$?
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All real numbers. Logarithmic functions output all real values.
All real numbers. Logarithmic functions output all real values.
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State the property: $\text{log}_b(xy)$.
State the property: $\text{log}_b(xy)$.
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$\text{log}_b(x) + \text{log}_b(y)$. Product rule: logarithm of product equals sum of logarithms.
$\text{log}_b(x) + \text{log}_b(y)$. Product rule: logarithm of product equals sum of logarithms.
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What is the inverse of an exponential function $y = b^x$?
What is the inverse of an exponential function $y = b^x$?
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$x = \text{log}_b(y)$. Exponential and logarithmic functions are inverse operations.
$x = \text{log}_b(y)$. Exponential and logarithmic functions are inverse operations.
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What is $\text{log}_b(1)$ for any base $b$?
What is $\text{log}_b(1)$ for any base $b$?
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- Any base raised to power 0 equals 1.
- Any base raised to power 0 equals 1.
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What is the range of $f(x) = b^x$ for $b > 1$?
What is the range of $f(x) = b^x$ for $b > 1$?
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$(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.
$(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.
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Solve for $x$: $e^x = 1$.
Solve for $x$: $e^x = 1$.
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$x = 0$. Any number to the power of 0 equals 1.
$x = 0$. Any number to the power of 0 equals 1.
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State the property: $\text{log}_b(x^r)$.
State the property: $\text{log}_b(x^r)$.
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$r \text{log}_b(x)$. Power rule: exponent becomes coefficient in logarithm.
$r \text{log}_b(x)$. Power rule: exponent becomes coefficient in logarithm.
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