Can I use ln instead of log in the change of base formula?

Yes. The formula works with any base a. ln(x) ÷ ln(b) gives the same result as log(x) ÷ log(b). Just keep the numerator and denominator in the same base.

What if my calculator does not have a log button at all?

Use a four-figure log table. Find log₁₀ of the argument and the base in the table, then divide by hand. The characteristic comes from the number's order of magnitude.

Why does log₂(8) equal 3 but log(8) ÷ log(2) gives 0.903?

It does not. log(8) ÷ log(2) is 0.903 ÷ 0.301 = 3. You likely typed log(2) ÷ log(8) instead. Check the division order.

Does the change of base formula work for negative arguments?

No. Real logarithms are undefined for x ≤ 0. The TI-84 returns ERR:DOMAIN for value ≤ 0. The domain of a logarithm is positive real numbers only.

Change of Base Formula for Logarithms

The Change of Base Formula

Your calculator has a log button for base 10 and an ln button for base e, but the problem uses base 3, base 7, or base 2. The change of base formula lets you rewrite a logarithm in any base as a ratio of logs in a base your calculator does support.

log_b(x) = log_a(x) / log_a(b)

Choose the new base a as 10 or e, whichever your calculator gives you. The result is the same either way. To evaluate log₃(20), press log(20) ÷ log(3). The result is about 2.727.

Short Proof of the Change of Base Formula

The proof takes two lines if you know the power and product properties. Start with the definition: log_b(x) = y means b^y = x. Take log_a of both sides: log_a(b^y) = log_a(x). Use the power rule: y · log_a(b) = log_a(x). Solve for y: y = log_a(x) / log_a(b). That is the formula.

No case is special. The same proof works whether you use base 10, base e, or base 2 as your a. The only requirement is that a, b, and x are positive and b ≠ 1.

Change of Base Logarithm Examples

Example 1: Base 3, Argument 20

Evaluate log₃(20). Use base 10 on a calculator: log(20) ÷ log(3). log(20) ≈ 1.3010. log(3) ≈ 0.4771. 1.3010 ÷ 0.4771 ≈ 2.727. Check: 3^2.727 ≈ 20, correct.

Example 2: Base 2, Argument 50

Evaluate log₂(50). A common computer science binary logarithm. log(50) ÷ log(2) ≈ 1.6990 ÷ 0.3010 ≈ 5.644. You can also use natural logs: ln(50) ÷ ln(2) ≈ 3.9120 ÷ 0.6931 ≈ 5.644. Same result. This is the binary log used in algorithm complexity analysis in texts like CLRS.

Example 3: Base 7, Argument 0.5

Evaluate log₇(0.5). The argument is less than 1, so the answer will be negative. log(0.5) ÷ log(7) ≈ (−0.3010) ÷ 0.8451 ≈ −0.3562. Check: 7^−0.3562 ≈ 0.5, correct. The same method works for any positive argument.

How to Do Log Base on TI-84 and Phone Calculators

On a TI-84 Plus CE

The logBASE( function is a dedicated tool that removes the need to divide two logs. Press the MATH key, scroll down to option A (logBASE(), and press ENTER. Enter the value, then a comma, then the base. For log₃(20): press MATH, A, then type 20 , 3 ) and ENTER. The TI-84 Plus CE guidebook (Texas Instruments, 2023) confirms the syntax is logBASE(value, base). The base is required; omitting it triggers an ERR:DOMAIN.

On a Phone Calculator App

Most scientific calculators lack a dedicated logBASE key. Use the change of base formula manually: type the log of the argument, press ÷, then the log of the base. On the iPhone calculator in landscape mode, the log button is base 10; the ln button is base e. Both work. Paste the full expression, log(20)÷log(3), into the entry line to avoid rounding intermediate steps.

Common Mistake: Dividing the Wrong Way

The most frequent error is reversing the numerator and denominator. log_b(x) = log(x) ÷ log(b), not log(b) ÷ log(x). If you type log(3) ÷ log(20) for log₃(20), you get 0.367 instead of 2.727, a factor of about 7.4 off.

A second common mistake is treating log(x) as a variable or assuming log(x) and ln(x) are interchangeable in the same formula. They are different functions: log is base 10, ln is base e. Both work in the change of base formula, but you must use the same base for both the numerator and denominator. Mixing log in the numerator and ln in the denominator gives a wrong result by a factor of about 2.3026.