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
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 |
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 |
Check: | |
| Substitute the solution, 13, in place of x |
| Simplify |
2 = 2 | |
| |
| Property of Logarithms: |
| Definition of Logarithm |
8 = x | Simplify |
0 = x | 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 |
| | ||
| Substitute the solution 8 for x | | Substitute the solution –1 for x |
| Subtract | | Subtract |
3 + 0 = 3 | | The number -1 does not check, since negative numbers do not have logarithms | |
3 = 3 |
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 | Distributive Property |
2x | 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 = |
log (2x – 1) = log (4x – 3) – log x | |
| Substitute the solution |
log 2 = log 3 – log 1.5 | Simplify |
log 2 = log | Property of Logarithms: |
log 2 = log 2 |
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 |