AP Precalculus: Logarithmic Functions

On a semi-log plot the x-axis is ordinary, but the y-axis is logarithmic: the spacing of tick marks depends on log y, so each step up multiplies the value instead of adding to it. This lets you display data that cover a huge range of values. The key fact is that an exponential function becomes a straight line on a semi-log plot. To see why, take a log of both sides of y = a·b^x: log y = x·log b + log a. This is a line with slope log b and intercept log a. Similarly, for y = a·e^kx, ln y = kx + ln a. This process is called linearizing. If data points fall on a line on a semi-log plot, an exponential model fits well, and the line’s slope and intercept give you b and a.

1. Logarithmic Expressions and Evaluating Logarithms

A logarithm answers one question: what exponent do I put on the base to get this number? The expression log_b c, read “log base b of c,” is the exponent a that makes b^a = c. The two statements are the same fact in two forms: log_b c = a means b^a = c, where b > 0, b ≠ 1, and c > 0. When the base is omitted, as in log x, it is understood to be 10 (a common logarithm), so log 1,000 = 3 because 10³ = 1,000. Some values can be found by thinking in powers, and others need a calculator. If your calculator only has the LOG (base 10) and LN (base e) keys, use the change of base formula log_b x = log x / log b (or ln x / ln b, which gives the same result). A logarithm of a number between 0 and 1 is negative, log_b 1 = 0 for every base, and log_b b = 1.

2. Logarithms as Inverses of Exponentials

The logarithmic function y = log_b x is defined as the inverse of the exponential function y = b^x, so y = log_b x means b^y = x. Because they are inverses, each one undoes the other. This gives the two identities you will use constantly: log_b(b^x) = x for every real x, and b^{log_b x} = x for every x > 0. In words, the log “cancels” a power of the same base, and a power “cancels” a log of the same base. The identities also hold for common logs (10^{log x} = x, log 10^x = x) and natural logs (e^{ln x} = x, ln e^x = x). Watch the base: the identity only works when the base of the log and the base of the power match.

3. Graphs of Logarithmic Functions: Domain, Range, and Key Features

Because y = log_b x is the inverse of y = b^x, its graph is the reflection of the exponential graph over the line y = x, and every ordered pair (s, t) of the exponential becomes (t, s) for the logarithm. So to graph y = log_b x, make a table for b^x and swap the columns. The inputs and outputs swap roles, so the domain and range swap too. Summary of y = log_b x:

• Domain: x > 0, that is (0, ∞). Range: all real numbers.
• x-intercept (1, 0), because b⁰ = 1. There is no y-intercept.
• Vertical asymptote: x = 0 (the y-axis).
• If b > 1 the graph rises as x increases. If 0 < b < 1 it falls as x increases.
• It lives in Quadrants I and IV, is always increasing or always decreasing, and is always concave down (b > 1) or concave up (0 < b < 1), so it has no extrema or points of inflection.

Compare with y = b^x: domain all reals, range y > 0, y-intercept (0, 1), horizontal asymptote y = 0.

4. End Behavior and Limits of Logarithmic Functions

Logarithmic functions have the vertical asymptote x = 0 and an unbounded end behavior, so every limit at the ends is ∞ or −∞. For y = log_b x with b > 1: lim_x→∞ = ∞, lim_x→0⁺ = −∞, and lim_x→1 = 0. With 0 < b < 1 the graph is flipped, so the two infinite limits swap signs. A multiplier a in y = a·log_b x also flips the signs when a is negative. Only the right-hand limit x → 0⁺ exists, because the function is not defined for x ≤ 0. Although the graph keeps rising, it does so very slowly.

5. Laws of Logarithms

Because a logarithm is an exponent, the logarithm laws mirror the exponent laws. For positive x and y and any valid base:

• Product: log_b(xy) = log_b x + log_b y
• Quotient: log_b(x/y) = log_b x − log_b y
• Power: log_b(xⁿ) = n·log_b x, which also covers roots: log_b √x = (1/2) log_b x
• Change of base: log_b x = log_a x / log_a b

There are two common tasks. To expand, break a single log into a sum and difference of simple logs. To condense, combine sums and differences into one log. If you are given the values of a few logs and asked for another, factor the number into pieces whose logs you know. Be careful: there is no rule that simplifies log(x + y) or (log x)/(log y).

6. Using the Laws to Describe Transformations; the Natural Logarithm

The laws explain how the graph of a log function is transformed. Every horizontal dilation becomes a vertical translation, because log_b(kx) = log_b k + log_b x. Raising the input to a power becomes a vertical dilation, since log_b(xⁿ) = n log_b x. And by change of base, all logarithmic functions are vertical dilations of each other: log_b x = (1/log_a b)·log_a x. To describe a transformation, first rewrite the function as a·log_b x + k in the same parent base, then read off the dilation and shift. The natural logarithm ln x is log_e x, with e ≈ 2.718. It has the same laws, plus ln e = 1, ln e^x = x, and e^{ln x} = x. Use the LN key for values such as ln 5 ≈ 1.6094.

7. Solving Logarithmic and Exponential Equations

Logarithmic equations. If every term is a log with the same base, use the laws to write each side as a single log, then use the one-to-one property: if log_b A = log_b B, then A = B. If the equation has a constant, collect all the logs on one side, condense into one log, rewrite in exponential form, and solve. Always check every answer in the original equation and reject any that puts a zero or negative number inside a log; those are extraneous solutions. Exponential equations. If both sides cannot be written with the same base, take a log of both sides and use the power law to bring the variable down: from b^kx = c get kx·log b = log c, so x = log c / (k log b). For models such as f(t) = a·e^kt, take ln of both sides.

8. Exponential and Logarithmic Inequalities

The methods for equations carry over to inequalities, with two extra cautions. First, the domain: inputs of a log must be positive, so always intersect your answer with the domain. Second, the direction of the base: if the base is greater than 1 the function is increasing, so the inequality direction is kept when you drop the logs or rewrite in exponential form. If the base is between 0 and 1 the function is decreasing, so the inequality reverses. A reliable plan is to solve the related equation first, then use the domain and a test value in each region (or the graph) to decide which side to keep.

9. Transformed Logarithmic Functions, Domains, and Inverses

A transformed logarithm has the form f(x) = a·log_b(x − h) + k. The horizontal shift moves the vertical asymptote to x = h and the domain to x > h, and k shifts the graph vertically. The easiest way to graph it is to track the parent’s anchor point (1, 0): it moves to (1 + h, k). Reflection and dilations come from a. To find the domain of any log expression, set the input of the log greater than 0. To find the inverse of a transformed function, swap x and y, isolate the log (or the exponential), rewrite in the other form, and solve for y. The inverse of a log is an exponential and vice versa, and the asymptote of one becomes the asymptote of the other, rotated (vertical ↔ horizontal).

10. Logarithmic Models and Regression

Logarithmic functions model quantities that grow quickly at first and then slow down but never stop increasing, such as a learning curve, or any quantity where equal ratios of the input give equal differences in the output (the Richter and decibel scales, for example, are logarithmic because the intensity is an exponential function of the reading). To build a model y = a + b ln x (or a·log_b x + k) from points, use a point with x = 1, where ln 1 = 0, to find the constant a, and then a second point to find b. With data and a calculator, use logarithmic regression (LnReg), which returns a and b for y = a + b ln x. Using a model inside the range of the data is interpolation and is generally reliable. Using it outside the range is extrapolation and is much less certain.

11. Semi-Log Plots and Linearizing Exponential Data

On a semi-log plot the x-axis is ordinary, but the y-axis is logarithmic: the spacing of tick marks depends on log y, so each step up multiplies the value instead of adding to it. This lets you display data that cover a huge range of values. The key fact is that an exponential function becomes a straight line on a semi-log plot. To see why, take a log of both sides of y = a·b^x: log y = x·log b + log a. This is a line with slope log b and intercept log a. Similarly, for y = a·e^kx, ln y = kx + ln a. This process is called linearizing. If data points fall on a line on a semi-log plot, an exponential model fits well, and the line’s slope and intercept give you b and a.

Common Mistakes to Avoid

• Mixing up the forms. log_b c = a means b^a = c. The base stays the base and the log is the exponent.
• Splitting a log of a sum. log(x + y) is not log x + log y. The product law applies to log(xy), not to a sum inside the log.
• Forgetting the domain. The input of a log must be positive. Always check solutions in the original equation and reject extraneous ones.
• Dividing logs when you should subtract. log(x/y) = log x − log y, which is not (log x)/(log y).
• Moving a coefficient the wrong way. 3 log x = log x³, not log(3x).
• Using the wrong base in the identities. b^{log_b x} = x only when both bases match.
• Ignoring a base between 0 and 1 in inequalities. A logarithm with base less than 1 is decreasing, so the inequality direction reverses.

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