DPST1012 Chap.7 Exponentials, Logarithms and Inverse Functions
Exponentials, Logarithms and Inverse Functions
Where this topic sits
Topic 3 runs in Week 4, the same week as the first 40-minute test, and it is examined in the Week 8 test and the final exam. The syllabus lists the laws of logarithms, solving exponential and logarithmic equations, the inverse function, sketching both families, the connection between them, and exponential and logarithmic inequalities.
Exponential models already appeared in Week 2 as the shape of rapid growth.
Inverses and the mirror in y = x
A function has an inverse only if it is one-to-one, the horizontal line test property from Week 2. The inverse undoes the function: it swaps inputs and outputs, so its domain is the original range and its range the original domain, and its graph is the reflection of the original in the line y = x.
The exponential a to the power x and the logarithm with base a are exactly such a pair. The exponential has domain ℝ, range above zero and a horizontal asymptote; the logarithm has domain above zero, range ℝ and a vertical asymptote.
Equations and inequalities
Exponential equations are solved by matching bases when possible and by taking logarithms otherwise, which brings the unknown down from the exponent.
Logarithmic equations are solved by combining logs with the product, quotient and power laws and then converting to exponential form, and every candidate must be checked because a logarithm only accepts positive inputs. Inequalities need one extra thought: a base between 0 and 1, or division by a negative logarithm, reverses the direction of the inequality.
What this chapter covers
- 01
Exponential functions and their graphs
- 02
Logarithms as inverse exponentials
- 03
Finding an inverse function
- 04
Laws of logarithms and change of base
- 05
Solving exponential equations
- 06
Solving logarithmic equations
- 07
Exponential and logarithmic inequalities
Worked example · free
Solve a log equation and an exponential inequality
- 1Quotient law: log₃((x + 6)/(x − 2)) = 2, so (x + 6)/(x − 2) = 3² = 9.
- 2x + 6 = 9x − 18 gives 8x = 24 and x = 3; both x + 6 and x − 2 are positive at x = 3, so it is valid.
- 10.5ᵗ < 0.05, so t ln 0.5 < ln 0.05.
- 1ln 0.5 is negative, so dividing reverses the sign: t > ln 0.05 / ln 0.5 ≈ 4.32.
Key terms
- Inverse Function
- The function that reverses a one-to-one function, sending each output back to the input that produced it.
- Logarithm
- The power to which a base must be raised to give a number, so that log base a of y equals x when a to the x equals y.
- Natural Logarithm
- The logarithm with base e, written ln, which is the inverse of the exponential function e to the x.
- Asymptote
- A line that a graph approaches more and more closely without reaching, such as y = 0 for an exponential.
- Change of Base
- The identity log base a of m equals ln m divided by ln a, used to evaluate any logarithm on a calculator.
Exponentials, Logarithms and Inverse Functions FAQ
Why can a logarithm not take a negative input?
Because a positive base raised to any real power is always positive, so no exponent produces a negative number or zero. The logarithm, which reverses that process, is therefore only defined for positive inputs.
When does an inequality flip in this topic?
When you multiply or divide by a negative number, which includes the logarithm of any number between 0 and 1. It also happens implicitly when an exponential base is below 1, because that function is decreasing.
How do I check that a function has an inverse?
Use the horizontal line test on its graph. If some horizontal line meets the graph twice, two inputs share an output and the function cannot be reversed unless its domain is restricted.
Is it better to use ln or log base 10 when solving?
Either works, as long as you use the same one on both sides. The natural logarithm is usual because it pairs with e and appears again in calculus, where its derivative is simply 1 over x.
Why does matching bases work for some exponential equations?
Because an exponential function with a base other than 1 is one-to-one, equal powers of the same base must have equal exponents. Writing 16 as 2 to the 4, for example, turns 2 to the x equals 16 into x equals 4 without logarithms.
Exam move
Sketch y = 2ˣ and its inverse on one grid from memory each week, and for every equation you solve, substitute the answer back into the original logs before you move on. Add every flipped inequality to a short list of traps you review before the Week 8 test.
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