# 6.6 Exponential and logarithmic equations

 Page 1 / 8
In this section, you will:
• Use like bases to solve exponential equations.
• Use logarithms to solve exponential equations.
• Use the definition of a logarithm to solve logarithmic equations.
• Use the one-to-one property of logarithms to solve logarithmic equations.
• Solve applied problems involving exponential and logarithmic equations.

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

## Using like bases to solve exponential equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers $\text{\hspace{0.17em}}b,$ $S,$ and $\text{\hspace{0.17em}}T,$ where ${b}^{S}={b}^{T}\text{\hspace{0.17em}}$ if and only if $\text{\hspace{0.17em}}S=T.$

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.

For example, consider the equation $\text{\hspace{0.17em}}{3}^{4x-7}=\frac{{3}^{2x}}{3}.\text{\hspace{0.17em}}$ To solve for $\text{\hspace{0.17em}}x,$ we use the division property of exponents to rewrite the right side so that both sides have the common base, $\text{\hspace{0.17em}}3.\text{\hspace{0.17em}}$ Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for $\text{\hspace{0.17em}}x:$

## Using the one-to-one property of exponential functions to solve exponential equations

For any algebraic expressions and any positive real number $\text{\hspace{0.17em}}b\ne 1,$

Given an exponential equation with the form $\text{\hspace{0.17em}}{b}^{S}={b}^{T},$ where $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ are algebraic expressions with an unknown, solve for the unknown.

1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form $\text{\hspace{0.17em}}{b}^{S}={b}^{T}.$
2. Use the one-to-one property to set the exponents equal.
3. Solve the resulting equation, $\text{\hspace{0.17em}}S=T,$ for the unknown.

## Solving an exponential equation with a common base

Solve $\text{\hspace{0.17em}}{2}^{x-1}={2}^{2x-4}.$

Solve $\text{\hspace{0.17em}}{5}^{2x}={5}^{3x+2}.$

$x=-2$

#### Questions & Answers

what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions
the polar co-ordinate of the point (-1, -1)
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
tanh`(x-iy) =A+iB, find A and B
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
yeah
Morosi
prime number?
Morosi
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1