Review of Exponential and Logarithmic Expressions and Functions

Exponential and Logarithmic Expressions :

An exponential expression is an expression involving an exponent. Typically when we refer to
exponential expressions, we do assume the base is a constant and the exponent is a variable, such as
a^x . a must be a non-negative number and preferably greater than 1. (Why is this, you might ask?
Because negative numbers don ’t play well with non-integer exponents since fractional exponents
represent roots and you can’t take square roots (and other even-indiced roots) of negative numbers.
Zero is not a good base because when raised to a negative power you put a zero in the denominator of a
fraction. Finally, 1 is discouraged since 1 to any power is still 1.)

A special base for an exponential expression in the number e (recall that e ≈ 2.71828 and is an irrational
number similar to pi). The expression e^x is called the natural exponential and e is the natural base. e^x
is very useful and important in calculus.

A logarithmic expression represents an exponent. The value of a logarithm is equal to the exponent
needed to raise the base to equal the input. (Read the last two statements over and over – they
represent the foundation of logarithms.) Logarithmic expressions are written in the form log_a x where a
would be the base of the logarithm and x is the input. (This logarithm would be read “the logarithm base
a of x”.)

Examples:

Two special logarithms:
Common logarithm – logarithm base 10
Natural logarithm – logarithm base e

Special properties of all logarithms :
log_a 1 = 0  because anything to the 0 power is 1
log_a a = 1 because anything to the first power is itself
log_a a^x = x because the base must be raised to the x power

Laws applied to logarithms include:

Important by-product – the change of base formula :

Just as addition and subtraction, multiplication and division, and powers and roots are all examples of
inverse operations – where one action undoes the effect of the other – exponentials and logarithms with
the same base are inverse operations of each other.

Thus because the composition (one operation inside another operation) of
inverse expressions should return the input of the inside operation.

Exponential Functions:
f (x) = a^x , a > 0, a ≠ 1 a is the base and is a constant

If a > 1, then the function is always increasing with a graph like (1) below.
If 0 < a < 1, then the function is always decreasing with a graph like (2) below.

Transformations can be applied to shift, reflect, or stretch/shrink the function.

Logarithmic Functions: f (x) = log_a x, a > 0, a ≠ 1 a is the base

If a > 1, then the function is always increasing with a graph like (3) below.
If 0 < a < 1, then the function is always decreasing with a graph like (4) below.

Transformations can be applied to shift, reflect, or stretch/shrink the function.

Generalized Rules for solving Exponential & Logarithmic Equations

Unless you have a special form that can be factored first , most exponential equations can be solved as
follows:

If there is just one exponential expression in the equation:
a: take all necessary steps to isolate the expression on one side
b: take the common or natural logarithm of both sides
- traditionally the common log is used, but I tend to take the natural log in order
to avoid the confusion that the common log (which looks like any other
logarithm except without a stated base) sometimes present… you should
always use common log with base 10 exponentials and natural log with base
e exponentials
c: use the laws of logarithms to rewrite the expression without exponents
d: solve for the variable

If there are two exponential expressions in the equation:
a: follow steps (b) and (c) above
b: since they will most likely be more than one variable term involved , follow steps
involving grouping and factoring to isolate the variable on one side
c: then solve for the variable

Exception to the previous case – if the bases of the two exponential expressions are equal, then
to solve you simply set the exponents equal to each other. If the two exponential expressions are
related by some power or root, use exponential laws to rewrite them so that you have equivalent
bases, then set the exponents equal to each other and solve.

Some exponential expressions can be factored similar to quadratics. Remember

Solving logarithmic equations can be divided into two broad categories – those that can be written where
a logarithm is equal to a number and those that can be written where logarithms of the same base are
equal to each other. Use the laws of logarithms to rewrite multiple logarithmic expressions as a single
log, then use the appropriate case:

If a logarithm is equal to a number, rewrite as an exponential and solve.

If two logarithms of equal base are equal to each other, set the inputs equal to each other and
solve.

With logarithmic equations (and some exponential equations), you must be aware of extraneous solutions
– “solutions” that appear to be appear to be correct but don’t actually work out in the original equation.

Homework problems: