Converting Exponential to Logarithmic Form

Write $2^5 = 32$ in logarithmic form.

Identify the base: The base is $2$ (the number being raised to a power) = Base = $2$

Identify the exponent: The exponent is $5$ = Exponent = $5$

Identify the result: The result is $32$ = Result = $32$

Write in log form: $\log_{\text{base}}(\text{result}) = \text{exponent}$: $\log_2(32) = 5$ = $\log_2(32) = 5$

Answer: $\log_2(32) = 5$

Converting Logarithmic to Exponential Form

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

Identify the base: The base is $3$ (the subscript of log) = Base = $3$

Identify the exponent: The exponent is $4$ (the result of the log) = Exponent = $4$

Identify the argument: The argument is $81$ (inside the log) = Argument = $81$

Write in exponential form: $\text{base}^{\text{exponent}} = \text{argument}$: $3^4 = 81$ = $3^4 = 81$

Answer: $3^4 = 81$

Evaluating a Common Logarithm

Evaluate $\log(1000)$.

Recognize the base: $\log$ without a subscript means base 10 = $\log_{10}(1000)$

Ask: 10 to what power equals 1000?: $10^? = 1000$ = Find the exponent

Calculate powers of 10: $10^1 = 10$, $10^2 = 100$, $10^3 = 1000$ = $10^3 = 1000$

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

Answer: $\log(1000) = 3$

Evaluating a Logarithm with a Different Base

Evaluate $\log_5(125)$.

Set up the question: 5 to what power equals 125? = $5^? = 125$

Calculate powers of 5: $5^1 = 5$, $5^2 = 25$, $5^3 = 125$ = $5^3 = 125$

State the answer: Since $5^3 = 125$, then $\log_5(125) = 3$ = $\log_5(125) = 3$

Answer: $\log_5(125) = 3$

Evaluating a Natural Logarithm

Evaluate $\ln(e^4)$.

Recognize that $\ln$ means $\log_e$: $\ln(e^4) = \log_e(e^4)$ = Base is $e$

Ask: $e$ to what power equals $e^4$?: $e^? = e^4$ = Find the exponent

The answer is the exponent itself: $e^4 = e^4$ when the exponent is $4$ = $\ln(e^4) = 4$

Answer: $\ln(e^4) = 4$

Logarithm of 1

Evaluate $\log_7(1)$.

Set up the question: 7 to what power equals 1? = $7^? = 1$

Recall: any number to the power 0 equals 1: $7^0 = 1$ = $7^0 = 1$

State the answer: Since $7^0 = 1$, then $\log_7(1) = 0$ = $\log_7(1) = 0$

Answer: $\log_7(1) = 0$ (The log of 1 is always 0, regardless of base!)

Mistake: Confusing the base and the argument

Why: In $\log_b(x)$, students often mix up which number is the base and which is the argument.

Correct: The base is the small subscript number ($b$). The argument is inside the parentheses ($x$). Memory tip: The base goes at the **b**ottom (subscript).

Mistake: Thinking $\log(x + y) = \log(x) + \log(y)$

Why: This is FALSE! Logarithms do not distribute over addition.

Correct: The correct rule is $\log(x \cdot y) = \log(x) + \log(y)$. Only multiplication inside the log becomes addition outside.

Mistake: Forgetting that $\log$ without a base means base 10

Why: Students sometimes assume no base means base $e$ or base 2.

Correct: $\log(x) = \log_{10}(x)$ (common log). For base $e$, we write $\ln(x)$.

Mistake: Trying to take the logarithm of a negative number or zero

Why: The argument of a logarithm must be positive. $\log(-5)$ and $\log(0)$ are undefined.

Correct: Always check that the argument is positive before evaluating. $\log_b(x)$ requires $x > 0$.

Earthquake Magnitude (Richter Scale)

The Richter scale measures earthquake intensity using logarithms. Each whole number increase represents a 10-fold increase in amplitude.

A magnitude 7 earthquake is $10^{7-5} = 100$ times more powerful than a magnitude 5 earthquake.

Sound Intensity (Decibels)

Sound intensity is measured in decibels (dB), which uses a logarithmic scale. The formula is $dB = 10 \log\left(\frac{I}{I_0}\right)$.

A sound that is 1000 times more intense than the reference level has intensity $10 \log(1000) = 10 \times 3 = 30$ dB.

Algorithm Complexity in Computer Science

Many efficient algorithms have logarithmic time complexity, written as $O(\log n)$. This is why binary search is so fast!

To find a word in a dictionary of 1024 pages using binary search, you need at most $\log_2(1024) = 10$ steps.

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

A: Because no real exponent makes a positive base equal zero or negative. For example, there's no $x$ where $10^x = 0$ or $10^x = -5$.

Q: What's the difference between $\log$ and $\ln$?

A: $\log$ typically means $\log_{10}$ (base 10), used in science and engineering. $\ln$ means $\log_e$ (base $e \approx 2.718$), used in calculus and natural growth/decay.

Q: Why is the logarithm of 1 always equal to 0?

A: Because any number raised to the power 0 equals 1. So $b^0 = 1$ means $\log_b(1) = 0$ for any valid base $b$.