Linear: Units are equally spaced, fixed increment. | Logarithmic: Units represent multiplicative change, power of the base.

$log(x)$: Base 10 | $ln(x)$: Base $e$ (Euler's number).

Exponential: $y = b^x$, rapid growth | Logarithmic: $y = log_b(x)$, slower growth, inverse of exponential.

$y = log_b(x)$: Vertical asymptote at $x = 0$, passes through (1, 0) | $y = b^x$: Horizontal asymptote at $y = 0$, passes through (0, 1)

$y = log_b(x)$: Domain is $(0, \infty)$ | $y = b^x$: Domain is $(-\infty, \infty)$

$y = log_b(x)$: Range is $(-\infty, \infty)$ | $y = b^x$: Range is $(0, \infty)$

$y = log_b(x)$: Approaches infinity at a decreasing rate | $y = b^x$: Approaches infinity at an increasing rate

$y = log(x)$: Derivative is $\frac{1}{x \cdot ln(10)}$ | $y = ln(x)$: Derivative is $\frac{1}{x}$

Linear: Suitable for data with evenly distributed values | Logarithmic: Suitable for data with values spanning several orders of magnitude

$y = log_b(x)$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$ | $y = b^x$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$

The exponent to which a base must be raised to produce a given number. If $b^a = c$, then $log_b(c) = a$.

The base of a common logarithm is 10. It is written as $log(c)$.

The base of a natural logarithm is $e$ (Euler's number, ≈ 2.71828). It is written as $ln(c)$.

The number you're taking the logarithm of.

The base is the number that is raised to a power to obtain the argument. It must be positive and not equal to 1.

A scale in which units represent a multiplicative change of the base, where each unit is a power of the base.

The exponent to which you raise the base to get the argument.

Euler's Number is the base of the natural logarithm, approximately equal to 2.71828.

The inverse function is $x = log_b(c)$

The argument of $log_b(c)$ is $c$.

Identify the base (2), the exponent (x), and the result (32). Rewrite as $log_2(32) = x$.

Ask: 'To what power must I raise 5 to get 25?' Since $5^2 = 25$, $log_5(25) = 2$.

Convert to exponential form: $3^4 = x$. Then, $x = 81$.

Convert the logarithmic equation to exponential form: $2^5 = x$. Simplify to find $x = 32$.

Recognize that the base is 10. Determine the power of 10 that equals 10000: $10^4 = 10000$. Therefore, $log(10000) = 4$.

Use the property that $ln(e^x) = x$. Therefore, $ln(e^7) = 7$.

Convert to exponential form: $b^y = x$.

Divide both sides by 2: $log(x) = 3$. Convert to exponential form: $10^3 = x$. Therefore, $x = 1000$.

Convert to exponential form: $2^3 = x + 1$. Simplify: $8 = x + 1$. Subtract 1 from both sides: $x = 7$.

Convert to exponential form: $e^0 = x$. Since any number raised to the power of 0 is 1, $x = 1$.