Solving Exponentials Using Logarithms

Solving Exponentials!

Let's use logarithms to solve equations!

Now that we know WHAT a logarithm is, how does it help us?

Sometimes we are given a problem where x is in the exponent and we'll need to use logarithms to solve for x.

See how we can use the inverse of -2 to cancel the +2?

What next??

• Now we have to "un-do" the exponent

• We now know the inverse of exponent is logarithm!!

• So we "take the log" of both sides (I know, super fancy sounding!!)

• THEN, x-5 is no longer an exponent, and can move to the front of the equation!

• NOW, we can continue solving!

Multiple Select

One more...

$$7^{\left(x+3\right)}=46$$  

• $$x=\log\left(39\right)-3$$  

• $$x=\frac{\log\left(46\right)}{\log\left(7\right)-3}$$  

• $$x=\frac{\log\left(46\right)}{\log\left(7\right)}-3$$  

• $$x=\log_746-3$$  

Wait... There were two answers?

We can write our answer in multiple ways!!

Wait... What was that log₇46??

We need a way to go back and forth between base 10 and other bases.

This is called the CHANGE OF BASE formula!

Using Desmos

Desmos will evaluate logarithms, whether they simplify to whole numbers or not!

This also shows you WHERE to find the log buttons in the Desmos keyboard! (Or you can type it out as log)

Multiple Choice

Solve this one!

$$3\cdot4^{x-1}=24$$  

• $$x=1$$  

• $$x=\frac{\log8}{\log4}+1$$  

• $$x=\frac{\log21}{\log4}+1$$  

• $$x=1.5$$  

This is how the last problem should be solved!!

Multiple Choice

Solve:

$$5(6)^{3x}=20$$  

• 2.3

• .26

• .04

• 4

Multiple Choice

3^{2x – 6}  = 81

• x = log 4
• x = 5
• x = 4
• x = -1

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

• -7.374
• -8.643
• -7.360
• -8.853

Multiple Choice

Solve for x. 

$$3^{\left(n-7\right)}-3=24$$  

• 3

• 8

• 10

• No solution

Solving Exponential Equations

• Second Method: When there is an exponent on only ONE side.

Multiple Choice

3^{2x – 6}  = 81

• x = log 4
• x = 5
• x = 4
• x = -1

Multiple Choice

7ⁿ⁺¹⁰- 8 = 6

• -7.374
• -8.643
• -7.360
• -8.853

Multiple Choice

Solve the equation: 

$$6^{2x}=6^{\left(-x-1\right)}$$  

• $$-\frac{1}{3}$$  

• $$\frac{15}{8}$$  

• $$5$$  

• $$\frac{19}{10}$$  

Solving exponential equations

• Solve, giving your answers correct to 3 significant figures.

  5^2x+1=7

• log  5^2x+1=log  7

• (2x+1)log  5=log  7

• 2x  log  5+log  5=log  7

• x=log  7−log  5/2  log  5

• x=0.105

Example 3

• Solve, giving your answers correct to 3 significant figures.

  3^2x = 4^x+5

• log  3^2x = log  4^x+5

• 2x  log  3 = (x+5)log  4

• 2x  log  3 = x  log  4+5  log  4

• x(2  log  3−log  4)=5  log  4

• x=5  log  4/2  log  3−log  4

• x=8.55

Reminder on how to solve

• Isolate the base

• Take the Logarithm of both sides

• Bring the exponent to the front

• Continue solving!