Solving exponential equations

Notice the bases are not the same so you cannot set the exponents equal yet

Look at the expressions and see if you can rewrite them so the bases are the same

In the beginning it will typically mean rewriting the larger base in terms of the smaller base

 $3^x=9^4$  

 $3^x=\left(3^2\right)^4$  

Simplify the right side by using the power to a power property

 $3^x=3^8$  

Now you can set the exponents equal since the bases are the same

x = 8

 $2^{5x}=\left(2^2\right)^{2x-1^{ }}$  

Be careful with your power to a power property.  Here its 2(2x - 1) = 4x - 2

 $2^{5x}=2^{4x-2^{ }}$  

5x = 4x - 2

x = -2

 $4^x=64^3$  

 $4^x=\left(4^3\right)^3$  

 $4^x=4^9$  

x = 9

 $3^{2x}=9^{5x-4^{ }}$  

 $3^{2x}=\left(3^2\right)^{5x-4^{ }}$  

 $3^{2x}=3^{10x-8^{ }}$  

2x = 10x - 8

-8x = -8

x = 1

 $4^{2x+6^{ }}=64^{2x-4^{ }}$  

 $4^{2x+6^{ }}=\left(4^3\right)^{2x-4^{ }}$  

 $4^{2x+6^{ }}=4^{6x-12^{ }}$  

2x + 6 = 6x - 12

18 = 4x

 $x\ =\ \frac{9}{2}$  

Here you will need to rewrite both sides of the equation

 $\left(2^6\right)^{c-2^{ }}=\left(2^5\right)^{2c}$  

 $2^{6c-12^{ }}=2^{10c}$  

6c - 12 = 10c

-12 = 4c

c = -3

Remember your negative exponent properties?

 $\frac{1}{625}=625^{-1}$  

 $5^{3-2x^{ }}=625^{-1}$  

 $5^{3-2x^{ }}=\left(5^4\right)^{-1}$  

 $5^{3-2x^{ }}=5^{-4}$  

3 - 2x = -4

-2x = -7

 $x\ =\ \frac{7}{2}$  

A. Write the equation to model teen spending.

 $A\left(t\right)=79.7\left(1+.035\right)^t$  in billion $

 $A\left(t\right)=79.7\left(1.035\right)^t$  

B.  Find the expected amount of money teens will spend in 2021.  Round to the nearest hundredth

 $A\left(15\right)=79.7\left(1.035\right)^{15}$  

 = $133.53 billion 

Known: initial amount a = 1,321,045, final amount A(t) = 1,512,986, and time periods t = 9

Unknown: growth rate 1 + r

a = 1,321,045, A(t) = 1,512,986, and t = 9

 $A\left(t\right)=a\left(1+r\right)^t$  

 $1,512,986=1,321,045\left(1+r\right)^9$  

Solve for 1 + r

 $\frac{1,512,986}{1,321,045}=\left(1\ +\ r\right)^9$  

 $\left(\frac{1,512,986}{1,321,045}\right)^{\frac{1}{9}}=\left(\left(1+r\right)^9\right)^{\frac{1}{9}}$  

Rounded to the nearest tenth 1 + r = 1.02 which makes the model  $A\left(t\right)=1,321,045\left(1.02\right)^t$  

 $17942=9426\left(1+r\right)^7$  

 $\frac{17942}{9426}=\left(1+r\right)^7$  

 $\left(\frac{17942}{9426}\right)^{\frac{1}{7}}=\left(\left(1+r\right)^7\right)^{\frac{1}{7}}$  

1 + r = 1.0963

 $A\left(t\right)=9426\left(1.0963\right)^x$  

Use the compound interest formula with p = 4000, r = .054, n = 4, t = 8

 $A\ =4000\left(1+\frac{.054}{4}\right)^{\left(4\cdot8\right)}$  

$6143.56

In Aleks Section 7.2 due tomorrow 11:59 pm

Wednesday we are meeting to work on Albert.io assignment SAT Prep: Equivalent expressions due Wednesday night 11:59 pm