9.3 Logarithms and Logarithmic Properties

9.3 Logarithms and Logarithmic Properties

A logarithm is an exponent. Table 11 pairs logarithmic equations with exponential ones because the two languages name the same relationship. This section is the translation: log_b a = c if and only if b^c = a, plus the product, quotient, and power rules you need to expand or condense, and the domain restrictions AAF will enforce when you later solve. Skills Insight band 276–300 specifically includes evaluating logarithmic expressions and equations.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

Definition: log_b a = c iff b^c = a

The statement

log_b a = c ⇔ b^c = a

means: the logarithm log_b a is the exponent you put on base b to get a.

• b is the base of the log (and of the matching exponential).
• a is the argument (the number you are “taking the log of”).
• c is the value of the log — an exponent.

Worked evaluations (Skills Insight 276–300)

log_2 8 = 3, because 2^3 = 8. That is the worked identity this section must make automatic. Ask “2 to what power is 8?” not “2 times what is 8?”

Negative values of the log are normal: they mean a negative exponent, which is a reciprocal, not a negative argument. The argument 1/25 is still positive.

Identities that fall out of the definition:

• log_b 1 = 0
• log_b b = 1
• log_b (b^k) = k
• b^{log_b a} = a for a > 0

The last two say that exponential and logarithm with the same base are inverse functions. Applying one undoes the other, the same way f and f^{−1} undo each other.

Common log and natural log, at recognition level

AAF expects you to recognize two named logs, not to compute messy decimals by hand.

On AAF, log without a base is base 10, not base 2. If a stem writes log_2, the 2 is required. ln is always base e. You do not need a numeric value of e ≈ 2.718 to evaluate ln(e^4); the inverse identity does the work.

A calculator icon may appear on some items (College Board calculator policy). Handheld calculators are not allowed except with an approved accommodation. Prefer exact rewrite: log_2 32 = 5 is better than a decimal approximation of ln 32 / ln 2.

Product, quotient, and power rules

Let every log below have the same base b > 0, b ≠ 1, and let arguments be positive.

These are the exponential laws rewritten. Because b^c · b^d = b^{c+d}, adding exponents (logs) multiplies the arguments.

Worked: expand log(xy^2)

Expand log(xy^2) (any legal base, including common log).

1. Product rule: log(xy^2) = log x + log(y^2).
2. Power rule on the second term: log(y^2) = 2 log y.
3. Result: log x + 2 log y.

That is the expansion this section must produce without hesitation. Traps:

• log x + log y^2 is not fully expanded (the power rule still applies).
• 2 log(xy) is log((xy)^2) = log(x^2 y^2), a different expression.
• log x · 2 log y multiplies logs; the product rule adds them.
• log(x + y^2) cannot be split at all. There is no sum rule of the form log(x + y) = log x + log y.

Condense the other direction: 2 log_5 m − log_5 n = log_5 (m^2 / n). Bring coefficients back as exponents, then combine a difference as a quotient.

Another expansion: ln(8 / w^3) = ln 8 − ln(w^3) = ln 8 − 3 ln w. If you want, ln 8 = ln(2^3) = 3 ln 2, so 3 ln 2 − 3 ln w = 3 ln(2/w).

Change of base, as an optional exact-form tool

If AAF leaves an answer as a log with an inconvenient base, change of base rewrites it in common or natural logs:

log_b a = (log a) / (log b) = (ln a) / (ln b)

Example: log_3 11 = ln 11 / ln 3 = log 11 / log 3. That is an exact form, not a rounded decimal. You will use it in the next section when 3^{4x} = 11 produces x = (log_3 11)/4, which is also x = ln 11 / (4 ln 3).

Change of base is optional when the original log already matches the options. Do not change base just to introduce a calculator. Do not write log_3 11 = log 3 / log 11 — that is the reciprocal, log_11 3.

Domain: a > 0, b > 0, b ≠ 1

The real logarithm log_b a requires:

• argument a > 0 (you cannot take a real log of 0 or of a negative);
• base b > 0 and b ≠ 1 (the same restrictions as an exponential base).

So log_2 (x − 3) needs x − 3 > 0, hence x > 3. That restriction is not optional decoration; it is why solving log equations in the next section includes a domain check. log(0) and log(−5) are not real. log_1 5 is not defined. log_{−2} 8 is not an AAF real-valued log.

The range of y = log_b x is all real numbers: logs can be negative, zero, or positive, as log_2 (1/8) = −3 already showed. Graphically, y = log_b x is the reflection of y = b^x across the line y = x. The exponential’s horizontal asymptote y = 0 becomes the log’s vertical asymptote x = 0. Domain and range swap, which is what inverses do.

Inverse pair in function notation

If f(x) = b^x, then f^{−1}(x) = log_b x. So f(3) = 8 would mean f^{−1}(8) = 3 when b^3 = 8. The functions chapter’s practice with inverse-function evaluation reappears here, and this time the inverse has a name: logarithm.

A compact checklist before any expansion or evaluation:

1. Convert to exponential if you are evaluating a single log.
2. Apply product / quotient / power only to products, quotients, and powers, never to sums.
3. Keep a > 0.
4. Common log is base 10; ln is base e.

Once those four are automatic, solving — same-base exponents, converting 3^{4x} = 11 into a log, and exponentiating a log equation — is the last section of this chapter.