Exponential Equations

An exponential equation is an equation containing a variable in an exponent. The following property of logarithms can be used to solve many exponential equations:

Property of Logarithms, Part 2

If x, y, and a are positive numbers, a ¹ 1, then

1. If x = y, then

2. If , then x = y.

Exponential equations can be solved by isolating the exponential expression; using Property of Logarithms, Part 2 to take the log of both sides; simplifying; and then solving for the variable.

Example. Solve:

Solution.

	Use Property of Logarithms, part 2, to take the log of both sides
x log 10 = log 5.71	Property of Logarithms:
x × 1 = log 5.71	Property of Logarithms:
x » 0.7566

Example. Solve:

Solution.

	Divide both sides by 7
	Use Property of Logarithms, Part 2, to take the log of both sides
	Property of Logarithms:
	ln e = 1
	Divide both sides by 3
x » 1.266

Example. Solve:

Solution.

	Use Property of Logarithms, Part 2, to take the log of both sides
(x + 2) ln 2 = (2x + 1) ln 3	Property of Logarithms:
x ln 2 + 2 ln 2 = 2x ln 3 + ln 3	Distributive Property
x ln 2 - 2x ln 3 = ln 3 – 2 ln 2	Isolate terms with the variable on one side of the equation
x(ln 2 – 2 ln 3) = ln 3 – 2 ln 2	Factor out the common factor, x
x » 0.191	Divide both sides by ln 2 – 2 ln 3

Logarithmic Equations

Logarithmic equations contain logarithmic expressions and constants. When one side of the equation contains a single logarithm and the other side contains a constant, the equation can be solved by rewriting the equation as an equivalent exponential equation using the definition of logarithm. For example,

	Definition of logarithm
16 = x + 3	Simplify
13 = x	Solve for x

All solutions of logarithmic equations must be checked, because negative numbers do not have logarithms:

Check:
	Substitute the solution, 13, in place of x
	Simplify
2 = 2	because

If one side of a logarithmic equation contains more than one logarithm, use properties of logarithms to condense them into a single logarithm. For example:

	Property of Logarithms:
	Definition of Logarithm
8 = x- 7x	Simplify
0 = x- 7x – 8	Write quadratic equation in standard form
0 = (x – 8)(x + 1)	Solve by factoring
x – 8 = 0 or x + 1 = 0 x = 8 or x = -1

Check:

	Substitute the solution 8 for x		Substitute the solution –1 for x
	Subtract		Subtract
3 + 0 = 3	because ; because		The number -1 does not check, since negative numbers do not have logarithms
3 = 3

The solution set is {8}.

In the next example, every term contains a logarithmic expression. We will solve this equation by using logarithmic properties to rewrite each side as a single logarithm. We then use Property of Logarithms, Part 2, and set the quantities equal to each other.

Example. Solve: log (2x – 1) = log (4x – 3) – log x

Solution.

log (2x – 1) = log (4x – 3) – log x
	Property of Logarithms:
	Property of Logarithms, Part 2
x(2x – 1) = 4x – 3	Multiply both sides by x
2x - x = 4x – 3	Distributive Property
2x - 5x + 3 = 0	Write the quadratic equation in standard form
(2x – 3)(x – 1) = 0	Solve by factoring
2x - 3 = 0 or x – 1 = 0
2x = 3 or x = 1 x =

Check: x =

log (2x – 1) = log (4x – 3) – log x
	Substitute the solution for x
log 2 = log 3 – log 1.5	Simplify
log 2 = log	Property of Logarithms:
log 2 = log 2

Check: x = 1

log (2x – 1) = log (4x – 3) – log x
log (2(1) – 1) = log (4(1) – 3) – log 1	Substitute the solution 1 in place of x
log 1 = log 1 – log 1
0 = 0

Both solutions check, so the solution set is .