Logarithms - Theory & Concepts

Logarithms

Logarithms are the inverse of exponentiation. They answer the question: "To what power must we raise a base to get a certain number?" They are essential for solving equations where the variable is in the exponent.

Definition of a Logarithm

For $b \gt 0$ and $b \neq 1$ , the logarithmic form and exponential form are equivalent. Observe the relationship between $y = b^x$ and $y = \log_b(x)$ . They are reflections across the line $y = x$ .

Fundamental Equivalence

The relationship between logarithmic and exponential forms.

\log_b(x) = y \iff b^y = x

Variables

Symbol	Description	Unit
	The base of the logarithm and exponent (b > 0, b ≠ 1)	-
	The argument of the logarithm, equal to the base raised to the power y	-
	The exponent, or the value of the logarithm	-

Common Logarithm

A logarithm with base $10$ , written as $\log(x)$ .

Natural Logarithm

A logarithm with base $e$ ( $\approx 2.718$ ), written as $\ln(x)$ .

Note

Interact with the explorer below to visualize the relationship between logarithmic and exponential functions.

Logarithms & Exponentials Explorer

Base (b)2

Domain: $x > 0$

Base: $b > 1$

Asymptote: The log curve approaches $x = 0$ (the y-axis) but never touches it, because $b^y \neq 0$ .

Identity: $b^{\log_b(x)} = x$

Logarithmic Function

y = \log_{2}(x)

Exponential Function

y = 2^{x}

Reflection Line

y = x

Properties of Logarithms

These properties allow us to expand or condense logarithmic expressions:

Product Rule: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$
Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$
Power Rule: $\log_b(x^p) = p \cdot \log_b(x)$
Change of Base: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

Note

Use the interactive sandbox below to experiment with the Laws of Logarithms. Select a rule and adjust the base and values to visually prove how expanding and condensing expressions yield the exact same numerical result.

Laws of Logarithms Sandbox

Select Logarithm Law

Base (b)10

Value x10

Value y5

Expanded vs Condensed Values

Condensed Left Side

\log_{10}(10 \cdot 5)

1.6990

Expanded Right Side

\log_{10}(10) + \log_{10}(5)

1.6990

Adjust sliders to watch both condensed and expanded mathematical equations update in real-time, verifying logarithmic parity.

Solving Exponential and Logarithmic Equations

The core strategy for solving these equations relies heavily on using inverse operations to isolate the variable.

Procedure

STEP-BY-STEP

For Exponential Equations (Variable in Exponent):

Isolate the exponential expression.
Take the logarithm (usually natural log, $\ln$ ) of both sides.
Use the Power Rule ( $\ln(x^p) = p \cdot \ln(x)$ ) to bring the variable down from the exponent.
Solve for the variable algebraically.

Procedure

STEP-BY-STEP

For Logarithmic Equations (Variable in Log Argument):

Condense the expression into a single logarithm using log properties (Product/Quotient rules).
Isolate the log expression.
Rewrite the equation in its equivalent exponential form (e.g., $\log_b(x) = y \implies b^y = x$ ).
Solve for the variable.
Check for extraneous solutions by ensuring your answer does not result in the log of a negative number or zero in the original equation.

Logarithmic Scale

A logarithmic scale is a way of displaying numerical data over a very wide range of values in a compact way. Each unit of distance on the scale represents a multiplication by the base, not an addition. It is often used for measuring the intensity of earthquakes (Richter scale), sound (decibels), or acidity (pH).

Logarithmic Scale Property

An increase of $1$ unit on a base- $10$ logarithmic scale means the measured quantity has multiplied by $10$ .

L = \log_{10} \left( \frac{I}{I_0} \right)

Note

Explore the logarithmic scale by interacting with the simulation below.

Linear vs. Logarithmic Scale

Energy / Intensity Factor (

10^x

)x = 3.0

1100M

Linear Scale

Value:

1.0k

Notice how the linear bar barely moves for small values, then shoots off completely for large ones.

Logarithmic Scale (Base 10)

$\\log_{10}(\\text{Value}):$

3.0

The logarithmic scale grows proportionally to the exponent, compressing huge ranges into manageable numbers.

Order of Magnitude: An increase of $+1$ on a Base-10 logarithmic scale means the underlying value gets 10 times larger. An increase of $+2$ means it gets 100 times larger.

Richter Scale

A magnitude 6.0 earthquake has 10 times the wave amplitude of a 5.0.

Decibels (dB)

Every 10 dB increase represents a 10-fold increase in sound intensity.

pH Scale

A pH of 4 has 10 times the hydrogen ion concentration of pH 5.

Key Takeaways

The Core Idea: $\log$ is just an exponent. $\log_{10}(100) = 2$ because $10^2 = 100$ .
Domain Restriction: You cannot take the logarithm of zero or a negative number.
Product/Sum Connection: Logs turn multiplication into addition and division into subtraction.
Calculators: Use the Change of Base formula if your calculator only has $\log$ (base $10$ ) and $\ln$ (base $e$ ).

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