Evaluating a Base-3 Logarithm with log

Evaluate $\log_3 50$. Round to 2 decimal places.

Write the change of base formula with base 10: $\log_3 50 = \dfrac{\log 50}{\log 3}$ = $\dfrac{\log 50}{\log 3}$

Find both common logarithms on the calculator: $\log 50 \approx 1.69897$ and $\log 3 \approx 0.47712$ = Keep all the digits for now

Divide, then round only at the end: $1.69897 \div 0.47712 \approx 3.56088$ = $\approx 3.56$

Check with the definition: $3^3 = 27$ and $3^4 = 81$, so the answer must lie between 3 and 4. Also $3^{3.56} \approx 50$ = Reasonable

Answer: $\log_3 50 \approx 3.56$

The Same Logarithm with ln

Evaluate $\log_3 50$ again, this time with natural logarithms. Round to 2 decimal places.

Write the formula with base $e$: $\log_3 50 = \dfrac{\ln 50}{\ln 3}$ = $\dfrac{\ln 50}{\ln 3}$

Find both natural logarithms: $\ln 50 \approx 3.91202$ and $\ln 3 \approx 1.09861$ = Different numbers than with log

Divide and round: $3.91202 \div 1.09861 \approx 3.56088$ = $\approx 3.56$

Answer: $\log_3 50 \approx 3.56$: the numerator and denominator change, but the quotient is the same. It does not matter whether you use log or ln.

An Exact Answer with a Clever Base

Find the exact value of $\log_4 8$.

Notice that 4 and 8 are both powers of 2: $4 = 2^2$ and $8 = 2^3$ = Choose the new base $c = 2$

Apply the change of base formula with base 2: $\log_4 8 = \dfrac{\log_2 8}{\log_2 4}$ = $\dfrac{\log_2 8}{\log_2 4}$

Evaluate the two easy logarithms: $\log_2 8 = 3$ and $\log_2 4 = 2$ = $\dfrac{3}{2}$

Check: $4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$ = Correct

Answer: $\log_4 8 = \dfrac{3}{2} = 1.5$ (exact, no rounding needed)

Swapping Base and Argument

You know that $\log_2 5 \approx 2.3219$. Find $\log_5 2$ without the log key. Round to 2 decimal places.

Write both logarithms with the change of base formula: $\log_2 5 = \dfrac{\log 5}{\log 2}$ and $\log_5 2 = \dfrac{\log 2}{\log 5}$ = The fractions are upside down versions of each other

Use the reciprocal rule: $\log_5 2 = \dfrac{1}{\log_2 5}$ = $\dfrac{1}{2.3219}$

Divide and round: $1 \div 2.3219 \approx 0.43068$ = $\approx 0.43$

Answer: $\log_5 2 \approx 0.43$. Check: $2.3219 \times 0.43068 \approx 1$

Three Decimal Places and a Surprise

Evaluate $\log_8 1000$ with natural logarithms. Round to 3 decimal places, then compare with $\log_2 10$.

Write the formula with ln: $\log_8 1000 = \dfrac{\ln 1000}{\ln 8}$ = $\dfrac{\ln 1000}{\ln 8}$

Find both natural logarithms: $\ln 1000 \approx 6.907755$ and $\ln 8 \approx 2.079442$ = Keep all the digits

Divide and round to 3 decimal places: $6.907755 \div 2.079442 \approx 3.321928$ = $\approx 3.322$

Compare with $\log_2 10$: $\log_2 10 = \dfrac{\ln 10}{\ln 2} \approx 3.322$ as well, because $\ln 1000 = 3 \ln 10$ and $\ln 8 = 3 \ln 2$, so the 3s cancel = $\log_8 1000 = \log_2 10$

Answer: $\log_8 1000 \approx 3.322$, exactly the same number as $\log_2 10$

Mistake: Writing the fraction upside down: $\log_3 50 = \dfrac{\log 3}{\log 50} \approx 0.28$

Why: The base goes in the denominator. A quick estimate shows the error: $3^3 = 27$ and $3^4 = 81$, so $\log_3 50$ must be between 3 and 4, not below 1.

Correct: Argument on top, base on the bottom: $\log_3 50 = \dfrac{\log 50}{\log 3} \approx 3.56$

Mistake: Turning the quotient of logs into the log of a quotient: $\dfrac{\log 50}{\log 3} = \log\dfrac{50}{3} \approx 1.22$

Why: The quotient rule says $\log 50 - \log 3 = \log \dfrac{50}{3}$. It is about a difference of logs, not a division of logs.

Correct: $\dfrac{\log 50}{\log 3}$ is a plain division of two numbers: $1.69897 \div 0.47712 \approx 3.56$

Mistake: Cancelling the word log: $\dfrac{\log 50}{\log 3} = \dfrac{50}{3} \approx 16.67$

Why: log is a function, not a factor. You cannot cancel it like a common number in a fraction.

Correct: Evaluate $\log 50$ and $\log 3$ first, then divide: $\approx 3.56$

Mistake: Rounding too early: $\dfrac{1.70}{0.48} \approx 3.54$

Why: Rounding $\log 50$ and $\log 3$ before dividing changes the second decimal place of the answer.

Correct: Divide with all the calculator digits (or type log(50)/log(3) in one go) and round only the final result: $3.56$

How Long Until Your Savings Double?

With yearly compound interest, money is multiplied by the same growth factor every year. The doubling time is a logarithm whose base is that factor.

At 3 % per year the factor is $1.03$, so the money doubles when $1.03^t = 2$, that is after $t = \log_{1.03} 2$ years.

Binary Search in Computer Science

Binary search finds an item in a sorted list by halving the list again and again. The number of halvings is a base-2 logarithm, and computers evaluate it with change of base.

A sorted list of 1024 names needs $\log_2 1024 = 10$ halvings, because $2^{10} = 1024$.

Bacteria That Triple Every Hour

Some bacteria cultures triple every hour. The time to reach a certain size is a base-3 logarithm.

After 2 hours the culture is $3^2 = 9$ times bigger, after 4 hours $3^4 = 81$ times bigger.

Q: Should I use log or ln in the formula?

A: Either one. $\dfrac{\log 50}{\log 3}$ and $\dfrac{\ln 50}{\ln 3}$ are both $\approx 3.56$. Just use the same kind of logarithm on top and on the bottom.

Q: My calculator has a $\log_b$ key. Do I still need the formula?

A: The key gives you the number, but the formula is still needed to find exact answers like $\log_4 8 = \dfrac{3}{2}$, to simplify expressions, to solve equations by hand and on tests where only log and ln are allowed. The calculator key itself uses change of base internally.

Q: How many decimal places should I round to?

A: As many as the question asks for. If it says nothing, 2 to 4 decimal places is usual (for example $\log_3 50 \approx 3.56$ or $3.5609$). Always round only once, at the very end.

Q: Why is $\log_b a \cdot \log_a b = 1$?

A: With change of base, $\log_b a = \dfrac{\log a}{\log b}$ and $\log_a b = \dfrac{\log b}{\log a}$. Multiplying the two fractions cancels everything and leaves $1$. So the two logarithms are reciprocals.