5.1 Exponential & Logarithmic Expressions

Exponential Functions

An exponential function has the form f(x) = a·b^x, where the variable sits in the exponent and the base b is a positive number other than 1. When b > 1 the function grows (e.g. 2^x: 2, 4, 8, 16…); when 0 < b < 1 it decays (e.g. (1/2)^x: 1/2, 1/4, 1/8…). Two facts ALEKS tests constantly: any nonzero base raised to the 0 power equals 1 (b^0 = 1), and a negative exponent means reciprocal (b^(−n) = 1/b^n). So 5^0 = 1 and 2^(−3) = 1/2^3 = 1/8.

The exponent rules you already know still apply: b^m·b^n = b^(m+n), b^m/b^n = b^(m−n), and (b^m)^n = b^(mn). For example, 3^4·3^2 = 3^6 = 729, and (2^3)^2 = 2^6 = 64.

What a Logarithm Means

A logarithm is the inverse of an exponential — it answers the question "what power do I raise the base to?" The definition is the single most useful fact in this whole section:

log_b(x) = y ⟺ b^y = x

Read it as: "log base b of x equals y" means "b to the y gives x." The log is the exponent. Some quick reads:

• log_2(8) = 3, because 2^3 = 8.
• log_10(1000) = 3, because 10^3 = 1000.
• log_5(1) = 0, because 5^0 = 1.
• log_3(1/9) = −2, because 3^(−2) = 1/9.

Two special bases get shorthand: log x with no base written means base 10 (the common log), and ln x means base e (the natural log, covered below).

Converting Between Exponential and Log Form

This is a guaranteed ALEKS skill. To convert, match base, exponent, and result.

Worked example. Write 4^3 = 64 in log form. The base stays the base, the exponent becomes the log's value, and the result becomes the argument: log_4(64) = 3.

Worked example — the other direction. Write log_2(32) = 5 in exponential form. The base 2 raised to the value 5 gives the argument 32: 2^5 = 32. Always identify the base first, because that is the number being raised to a power and the one number that appears in both forms in the same role.

Why do logs even exist? Before calculators, multiplying large numbers was hard, but adding was easy — and logs convert multiplication into addition (the rules below). Today they matter because exponential and logarithmic functions describe compound interest, population growth, radioactive decay, pH, and earthquake (Richter) and sound (decibel) scales — anything that grows or shrinks by a constant factor rather than a constant amount.

The Three Log Rules

Because logs are exponents, the exponent rules translate into log rules that turn multiplication into addition:

1. Product rule: log_b(M·N) = log_b(M) + log_b(N)
2. Quotient rule: log_b(M/N) = log_b(M) − log_b(N)
3. Power rule: log_b(M^p) = p·log_b(M)

Worked example — expand. Expand log_2(8x³). Use product first, then power: log_2(8) + log_2(x³) = 3 + 3·log_2(x). (Here log_2(8) = 3.)

Worked example — condense. Write 2·log(x) + log(5) − log(y) as one log. Power rule turns 2·log(x) into log(x²); product and quotient combine the rest: log(5x²/y).

Worked example — numeric. Given log 2 ≈ 0.301 and log 3 ≈ 0.477, find log 6. Since 6 = 2·3, the product rule gives log 6 = log 2 + log 3 ≈ 0.301 + 0.477 = 0.778. And log 1.5 = log(3/2) = log 3 − log 2 ≈ 0.176.

There is also a change-of-base formula for evaluating any log on a calculator that only has base-10 or base-e keys: log_b(x) = log(x)/log(b) = ln(x)/ln(b). For example, log_2(10) = log(10)/log(2) = 1/0.301 ≈ 3.32, meaning 2^3.32 ≈ 10.

Natural Logs and e

The constant e ≈ 2.71828 is the natural base of growth, and ln x = log_e(x). The inverse identities are: ln(e^x) = x and e^(ln x) = x. So ln(e^4) = 4 and e^(ln 9) = 9 instantly. These appear when an equation has e in it.

Solving Exponential & Log Equations

Exponential equation — same base. Solve 2^(x+1) = 16. Write 16 = 2^4, so 2^(x+1) = 2^4 ⟹ x + 1 = 4 ⟹ x = 3.

Exponential equation — take a log. Solve 5^x = 40. Take log of both sides and use the power rule: x = log(40)/log(5) ≈ 1.602/0.699 ≈ 2.29.

Log equation — rewrite as exponential. Solve log_3(x) = 4. By the definition, x = 3^4 = 81.

Log equation with a rule. Solve log_2(x) + log_2(x−2) = 3. Condense: log_2(x(x−2)) = 3 ⟹ x(x−2) = 2^3 = 8 ⟹ x² − 2x − 8 = 0 ⟹ (x−4)(x+2)=0 ⟹ x = 4 or x = −2. Reject x = −2 (a log's argument must be positive), so x = 4.

Common mistake: log_b(M + N) is NOT log_b(M) + log_b(N). The product rule applies to multiplication inside the log, never addition. Also remember to discard solutions that make any log argument zero or negative — those are extraneous.