MATH1011 Chap.6 Exponential and Logarithmic Functions
Exponential and Logarithmic Functions
One inverse relationship, read in both directions
An exponential function takes a real exponent and returns a positive value. A logarithm reverses that action: log base a of y is the exponent placed on a to obtain y. This inverse relationship explains the shared points, the reflection of their graphs across the line y = x, and the way domain and range swap.
Exponentials accept every real input and stay positive; logarithms accept only positive inputs and can produce every real output.
The base must be positive and different from 1. When the base exceeds 1 the exponential graph increases; when it lies between 0 and 1 the graph decreases. In either case y = 0 is approached but never reached.
The logarithmic graph has the corresponding vertical asymptote at x = 0. These graph facts are not decorative: they tell you which candidate solutions are admissible before any detailed substitution.
Log laws are multiplication laws in disguise
The logarithm of a product becomes a sum, the logarithm of a quotient becomes a difference, and a power moves down as a multiplier.
None of these laws permits splitting a sum inside a logarithm. Write the argument conditions first, then combine. If a problem contains log(x − 1) and log(x + 3), the common admissible set is x greater than 1, not merely the domain of the expression obtained after multiplying the two arguments.
Changing base is another quotient law: log base a of x equals ln x divided by ln a.
It is especially useful when the given information uses common logarithms but the requested answer uses natural logarithms. Translate every named logarithm into the same base before combining coefficients.
Solving equations means filtering the algebraic answers
Combining logarithms often produces a quadratic equation, and that quadratic may have roots outside the original logarithmic domain.
Solve the algebra, then test each candidate against every original argument and against any variable base. A base must be positive, different from 1, and the argument must remain positive. Rejection is part of the solution, not an optional afterthought.
Exponential equations use the inverse move in the other direction. If both sides can be written with a common base, compare exponents.
Otherwise take logarithms after confirming both sides are positive. Keep exact logarithmic forms until the final line unless a decimal approximation is explicitly requested, because premature rounding can obscure whether two expressions are actually equal.
What this chapter covers
- 01
Exponential functions and their positive range
- 02
Logarithms as inverse functions
- 03
Product, quotient and power laws
- 04
Change of base and natural logarithms
- 05
Domain conditions for logarithmic equations
- 06
Candidate roots and rejection checks
Worked example · free
Solve a logarithmic equation and reject the impossible root
- 2State the common domain of the two logarithms.
- 3Combine the logarithms and solve the resulting quadratic.
- 2Filter the candidates through the original domain.
Key terms
- Exponential Function
- An Exponential Function places its variable in the exponent of a fixed admissible base.
- Logarithm
- A Logarithm returns the exponent required to produce a positive argument from its base.
- Natural Logarithm
- The Natural Logarithm is the logarithm whose base is the number e.
- Change of Base
- Change of Base rewrites a logarithm as a quotient of logarithms expressed in one convenient base.
- Admissible Base
- An Admissible Base for a real logarithm is positive and unequal to 1.
Exponential and Logarithmic Functions FAQ
Why must logarithmic roots be checked after solving?
Combining logs and solving the resulting algebraic equation can introduce candidates for which an original logarithmic argument is zero or negative. Those candidates satisfy the transformed equation but never belonged to the original domain, so they must be rejected explicitly.
Can I split log(x + y) into log x plus log y?
No. The addition rule applies to a product inside the logarithm, not to a sum. Log(xy) becomes log x plus log y, while log(x + y) normally stays intact. Confusing these shapes changes the function and usually creates false solutions.
How do exponential and logarithmic graphs help in an equation?
Their inverse relationship swaps coordinates and reflects the graphs across y = x. The exponential's positive range becomes the logarithm's positive-input domain, while the exponential's horizontal asymptote becomes the logarithm's vertical asymptote. Those constraints immediately rule out impossible inputs and outputs.
Exam move
Build a two-column sheet with each log law on the left and one illegal look-alike on the right. For every equation, reserve the first line for domain conditions and the last line for candidate checking. Practise moving between exponential and logarithmic form without a calculator, then use the question-bank answers to check exact forms rather than only decimals.
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