Logarithmic Function Context and Data Modeling - AP Precalculus

$x > 0$. Logarithm is undefined for non-positive values.

$2$. $4^2 = 16$, so the answer is 2.

$1$. $\text{ln}(e) = 1$ by definition of natural logarithm.

$x$. Exponential and natural log cancel as inverses.

$b = 10$. $10^2 = 100$, so base is 10.

$x$. Natural log and exponential cancel as inverses.

$0$. Any base raised to power 0 equals 1.

$y = 0$. Exponential functions approach zero as $x \to -\infty$.

$x = 0$. Logarithmic functions approach $-\infty$ as $x \to 0^+$.

$e$. Natural logarithm uses Euler's number as base.

$10$. Common logarithm uses base 10 by convention.

$-2$. $10^{-2} = 0.01$, so the answer is -2.

$b^y = x$. Logarithmic and exponential forms are equivalent.

$1$. Any base raised to the power 1 equals the base.

$3$. $3^3 = 27$, so the answer is 3.

All real numbers. Logarithmic functions can output any real value.

$x$. Natural log and exponential cancel as inverses.

$\text{log}_b(x) + \text{log}_b(y)$. Product rule: log of product equals sum of logs.

$y = 0$. Exponential functions approach zero as $x \to -\infty$.

$x = 0$. Logarithmic functions approach $-\infty$ as $x \to 0^+$.

$0$. Any base raised to power 0 equals 1.

$b^y = x$. Standard form showing logarithm-exponential relationship.

$-1$. Since $b^{-1} = \frac{1}{b}$, the log equals -1.

$5$. $2^5 = 32$, so the answer is 5.

$-\text{log}_b(x)$. Using the property $\text{log}_b(x^{-1}) = -\text{log}_b(x)$.

$b = 10$. $10^2 = 100$, so base is 10.

$3$. $3^3 = 27$, so the answer is 3.

$e^y = x$. Natural log uses base $e$ in exponential form.

$y$. By definition of logarithm from exponential form.