5.2 Logarithms and Solving Exponential Equations

What a logarithm means

A logarithm is an exponent question. The statement log_b(y) = x means b^x = y. Read it as, "base b raised to what exponent gives y?" This inverse relationship is included on the Mathematics 30-2 formula sheet, and it is why logarithms appear beside exponential functions in the Relations and Functions part of the diploma. When an exponential equation has a variable in the exponent, a logarithm can turn that exponent into a value you can calculate.

The base has restrictions: b must be positive and b cannot equal 1. The argument, which is the expression inside the logarithm, must also be positive. For log_3(x - 5), the domain is x > 5 because x - 5 must be greater than 0. For y = log_2(x) + 4, the vertical asymptote is x = 0; shifting inside the logarithm moves that boundary. These restrictions are not decoration. They decide which solutions are possible and which calculator answers must be rejected.

Rewriting between forms

The fastest way to evaluate simple logarithms is often to rewrite them in exponential form. If log_4(64) = x, then 4^x = 64, so x = 3. If log_5(1/25) = x, then 5^x = 1/25, so x = -2. A negative logarithm result is allowed; what cannot be negative is the argument of the logarithm.

Solving exponential equations

Use a common base when the numbers make it natural. For 3^(x+2) = 81, rewrite 81 as 3^4, then x + 2 = 4 and x = 2. For 2(5)^x = 250, divide first to get 5^x = 125, then 5^x = 5^3 and x = 3. This approach is exact and quick.

When a common base is not practical, use logarithms. Suppose 600(1.08)^t = 900. Divide by 600 to get (1.08)^t = 1.5. Taking logarithms gives t = log(1.5) / log(1.08), which is about 5.27 years. Natural logarithm and common logarithm both work if the same log type is used in the numerator and denominator. A graphing calculator can also solve by intersection, but the algebra explains why the answer is reasonable.

Log laws

Logarithm laws help simplify or solve, but only when their conditions are met. Product: log_b(MN) = log_b(M) + log_b(N). Quotient: log_b(M/N) = log_b(M) - log_b(N). Power: log_b(M^n) = n log_b(M). Change of base: log_b(M) = log(M) / log(b). The common diploma trap is inventing a law for addition, such as treating log_b(M + N) as log_b(M) + log_b(N). That is false. Test it with base 10: log(100 + 100) is log(200), not log(100) + log(100).

Worked example: domain and solution

Solve log_3(x - 1) = 4. First state the restriction: x - 1 > 0, so x > 1. Rewrite in exponential form: 3^4 = x - 1. Since 3^4 = 81, x = 82. The solution satisfies x > 1, so it is valid. A complete written response would show the restriction, the exponential rewrite, the calculation, and the final check.

Now solve 2^(x+1) = 19. A common base is not convenient. Take logarithms: x + 1 = log(19) / log(2), so x = log(19) / log(2) - 1, which is about 3.25. If a numerical-response item asks for the nearest hundredth, record 3.25, not the rounded intermediate ratio.

Diploma traps

The bulletin specifically notes difficulty with exponential equations that cannot be written as powers with a common base. Build a decision habit: simplify first, try common bases second, use logarithms or graph intersections third, then round according to the question. Keep the domain visible for logarithmic functions. If a calculator returns a value that makes a log argument zero or negative in the original equation, that value is not acceptable even if it appears after algebraic manipulation.

For context questions, a logarithmic answer often represents time, number of periods, or an input threshold. Do not leave the answer as "5.27" if the question asks when an investment reaches a target. Write "about 5.27 years" or, if the context requires whole years, explain whether the next full year is needed.

Calculator graphing checks

Graphing technology is acceptable for many Math 30-2 solving tasks, but the setup still matters. For 1.5(3)^x + 2 = 20, graph y1 = 1.5(3)^x + 2 and y2 = 20, or isolate first to get 3^x = 12. The logarithmic solution is x = log(12) / log(3), about 2.26. The graph should show the same intersection. If the algebra and graph do not agree, check whether you forgot to divide by the coefficient or subtract the vertical shift.

For logarithmic graphs, set a window that respects the vertical asymptote. A function such as y = log_2(x + 4) is undefined at x = -4 and to the left of that boundary. A calculator may simply refuse some points; your written work should explain the restriction rather than treating the blank graph as a technology problem.