Solving Exponentials Using Change of Base

Lesson 6.7 Solving Exponential Equations

Type 1: Same base on both sides

• Ignore the base

• Set the exponents equal to each other

• Solve

Multiple Choice

$$2^x=2^{\left(5x-80\right)}$$  

• $$x=-11$$  

• $$x=20$$  

• $$x=4$$  

• $$x=-16$$  

What if they don't have the same base?

• Can you change one of them to have the same base as the other?

Multiple Choice

$$7^3=49^{\left(x+5\right)}$$  

• $$x=5$$  

• $$x=-2$$  

• $$x=-\frac{7}{2}$$  

Type 2: Exponent only on 1 side

• Change to logarithmic form

• Use change of base formula to solve

Multiple Choice

$$10^x=1500$$  

• $$x=150$$  

• $$x=10$$  

• $$x=3.18$$  

Multiple Choice

$$5^{4x}=2115$$  

• $$x=19.03$$  

• $$x=1.19$$  

• $$x=4.78$$  

• $$x=12.98$$  

Multiple Choice

$$10^x+8=1008$$  

• $$x=10$$  

• $$x=3$$  

• $$x=8$$  

• $$x=100$$  

Multiple Choice

$$7^{\left(x+10\right)}-8=6$$  

• $$x=-7.374$$  

• $$x=-8.643$$  

• $$x=-7.360$$  

• $$x=-8.853$$