Logarithm Rules and How to Solve Exponential Equations

What a logarithm is

log_b(x) = y means exactly the same thing as b^y = x. The logarithm is the exponent. Since 10^3 = 1000, log_10(1000) = 3, and since 2^5 = 32, log_2(32) = 5.

The base b must be positive and not 1, and the argument x must be positive. log with no base written usually means base 10, and ln means the natural logarithm, base e, where e is about 2.718.

The three rules

Product rule: log_b(MN) = log_b(M) + log_b(N). Multiplying inside the log becomes adding outside.

Quotient rule: log_b(M/N) = log_b(M) - log_b(N). Dividing inside becomes subtracting outside.

Power rule: log_b(M^k) = k · log_b(M). An exponent inside comes out in front as a multiplier. This is the rule that solves exponential equations, because it brings the unknown down out of the exponent.

Each rule is an exponent law in disguise. The product rule, for example, is the log form of b^m · b^n = b^(m+n).

Change of base

Calculators have keys for log and ln but not for other bases. The change of base formula fixes that: log_b(x) = ln(x) / ln(b), and the same works with log in place of ln.

For example, log_3(20) = ln(20) / ln(3) ≈ 2.9957 / 1.0986 ≈ 2.7268.

Solving exponential equations

If both sides can be written as powers of the same base, set the exponents equal. In 5 · 2^x = 40, divide by 5 to get 2^x = 8 = 2^3, so x = 3.

If not, take the log of both sides and use the power rule. For 3^x = 20, ln(3^x) = ln(20) becomes x · ln(3) = ln(20), so x = ln(20) / ln(3) ≈ 2.7268. Check: 3^2.7268 is about 20.

Isolate the power first. In 4 · 3^x + 1 = 81, subtract 1 and divide by 4 before taking any logs: 3^x = 20, the equation just solved.

Solving log equations and checking for extraneous answers

Combine the logs into one, rewrite in exponential form, and solve. For log_2(x) + log_2(x - 2) = 3, the product rule gives log_2(x(x - 2)) = 3, so x(x - 2) = 2^3 = 8.

That becomes x^2 - 2x - 8 = 0, which factors as (x - 4)(x + 2) = 0, giving x = 4 or x = -2.

Now check both in the original equation. x = 4 gives log_2(4) + log_2(2) = 2 + 1 = 3, which works. x = -2 would need log_2(-2), which is undefined, so it is extraneous and the only solution is x = 4. Combining logs can introduce answers like this, which is why the check is not optional.

Mistakes that cost marks

log(a + b) is not log(a) + log(b). The product rule is about multiplying inside the log, not adding.

log(a) / log(b) is not log(a / b). The first is a change of base; the second is the quotient rule.

Mixing up log and ln on the calculator. Either works for change of base, as long as you use the same one on the top and bottom.

Taking the log before isolating the power, which leaves you with the log of a sum that the rules cannot split.

Calculator programs for this topic

These free TI-84 Plus CE Python programs carry out the steps above. Each page explains its own math, shows a screen preview, and lets you read the source before you download it.

• DuckieDai Logarithm Solver: Evaluates a log in any base, solves log_b(x) = y, and solves log_b(ax + c) = d with a check.
• DuckieDai Log and Exponent Solver: Finds the exponent, argument, or base, solves b^x = y, and converts between log and exponential form.