Converting Exponential to Logarithmic Form

Write $5^3 = 125$ in logarithmic form.

Identify the base: The base is $5$ (the number being raised to a power) = Base: $b = 5$

Identify the exponent: The exponent is $3$ = Exponent: $x = 3$

Identify the result: The result of $5^3$ is $125$ = Argument: $y = 125$

Write in logarithmic form: $\log_b(y) = x$ becomes $\log_5(125) = 3$ = $\log_5(125) = 3$

Answer: $\log_5(125) = 3$

Converting Logarithmic to Exponential Form

Write $\log_3(81) = 4$ in exponential form.

Identify the base of the logarithm: The subscript of $\log$ is $3$ = Base: $b = 3$

Identify the argument: The number inside the parentheses is $81$ = Argument: $y = 81$

Identify the logarithm value: The logarithm equals $4$ = Exponent: $x = 4$

Write in exponential form: $b^x = y$ becomes $3^4 = 81$ = $3^4 = 81$

Answer: $3^4 = 81$

Evaluating a Logarithm

Evaluate $\log_2(32)$.

Set up the question: We need to find $x$ where $2^x = 32$ = $2^x = 32$

Think about powers of 2: $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32$ = $2^5 = 32$

Identify the exponent: We need the exponent $5$ to get $32$ = $x = 5$

Answer: $\log_2(32) = 5$

Common Logarithm (Base 10)

Evaluate $\log(1000)$ (common logarithm, base 10).

Understand notation: $\log$ without a base means $\log_{10}$ (common logarithm) = $\log(1000) = \log_{10}(1000)$

Set up the equation: Find $x$ where $10^x = 1000$ = $10^x = 1000$

Express 1000 as a power of 10: $1000 = 10 \times 10 \times 10 = 10^3$ = $10^3 = 1000$

State the answer: Since $10^3 = 1000$, the exponent is $3$ = $\log(1000) = 3$

Answer: $\log(1000) = 3$

Mistake: Confusing the base and the argument

Why: In $\log_b(y) = x$, students sometimes swap $b$ and $y$. The base is the subscript number, the argument is inside parentheses.

Correct: Remember: $\log_{\text{base}}(\text{argument}) = \text{exponent}$. The BASE is what gets raised to a power.

Mistake: Thinking $\log_b(0)$ or $\log_b(-5)$ exists

Why: No positive base raised to any power can equal zero or a negative number.

Correct: The argument of a logarithm must always be positive. $\log_b(y)$ only exists when $y > 0$.

Mistake: Forgetting that $\log$ (no base written) means base 10

Why: The notation can be confusing. Some textbooks use $\lg$ for base 10.

Correct: $\log(x)$ is the common logarithm with base 10. $\ln(x)$ is the natural logarithm with base $e \approx 2.718$.

Mistake: Writing $\log_1(x)$

Why: $1$ raised to any power always equals $1$, so base $1$ cannot produce other values.

Correct: The base of a logarithm must be positive and not equal to $1$: $b > 0$ and $b \neq 1$.

Measuring Earthquake Magnitude

The Richter scale uses logarithms to measure earthquake intensity. Each whole number increase represents 10 times more ground motion.

A magnitude 6 earthquake has amplitude $A_6$, and magnitude 7 has amplitude $A_7 = 10 \times A_6$. The formula involves $\log_{10}$.

Calculating pH in Chemistry

The pH scale measures how acidic or basic a solution is using the formula $\text{pH} = -\log[H^+]$.

If $[H^+] = 0.001 = 10^{-3}$, then $\text{pH} = -\log(10^{-3}) = -(-3) = 3$ (acidic, like vinegar).

Q: Why is the base always positive and not equal to 1?

A: If the base were negative, raising it to fractional powers would give complex numbers. If the base were 1, then $1^x = 1$ for all $x$, so you could never get any other value. We need consistent, predictable results.

Q: What is the difference between log and ln?

A: $\log$ (common logarithm) uses base 10 and is convenient for decimal calculations. $\ln$ (natural logarithm) uses base $e \approx 2.718$ and appears naturally in calculus, growth/decay problems, and continuous compounding.

Q: Why can't we take the logarithm of zero or negative numbers?

A: No positive number raised to any real power equals zero or a negative number. For example, $2^x$ is always positive regardless of $x$. So $\log_2(0)$ and $\log_2(-5)$ have no real solutions.