Solving Log and Exponential Equations

We have solved simple log and exponential equations by setting either the exponents equal to each other or the pieces we are taking the logs of equal to each other. Nate is god

Solving Log and Exponential Equations

Example:log2(4) + log2(x) = log2(6) + log2(2)

Condense each side to a single log

log2(4x) = log2(6*2)log2(4x) = log2(12)

Set the parenthesis equal to each other4x = 12 4 4

x = 3

What happens if we can’t get a single log on each side?

log5(3x + 1) = 2

We need a new method to solve this

In order to solve any log function where we can’t get a single log on each side we need to use the following principle

log5(3x + 1) = 2

Remember that log functions and exponential functions are related to each other – you can change from one form to another.

So when solving a log function with a log on only one side you will solve by doing the following:

log5(3x + 1) = 2

1) First you will get a single log all by itself on one side of the equation and put back any exponents

2) Then convert the log form into the exponential form

3) You will now be able to solve the equation

Examples:4log3(x) = 28

328 = x4

Convert to exponential formlog3(x4) = 28

Put the Exponent Back

The calculator gives you scientific notation – that’s ok, just hit Ans^(1/4) to get the 4th root.

2187 = x

= x4( ( )1/4)1/4

Examples:1/3log2x + 5 = 7 Get the log by itself - 5 -51/3log2x = 2 Put the exponent back

log2(x1/3) = 2 Change to exponential form

22 = x1/3

4 = x1/3( ( )3)3

64 = x

Raise each side to the 3rd power (the opposite of the 1/3rd power)

Solving ln equations:16ln(x) = 30ln(x16) = 30

Put the exponent back

loge(x16) = 30

Change to loge form

e30 = x16 Enter e^30 in your calculator, then raise that answer to the 1/16th power to get the final answer

Change to exponential form

6.521 = x

Try These:6ln(4x) – 1 = 15

Put the exponent back

Change to loge formln(4x)6 = 16

Enter e^16 in your calculator, divide by 4096, then raise that answer to the 1/6th power to get a final answer.

Change to exponential form

e16 = (4x)6

Get the ln by itself +1 +16ln(4x) = 16

loge(4x)6 = 16

e16 = 46x6

e16 = 4096x6

3.046 = x

Solving exponential equations:10x + 5 = 60 - 5 - 5

Get the term with the exponent alone

10x = 55log1055 = x Use your calculator to get

the answer log(55)/log(10)

Change to log form

1.740 = x

Another Example:10-12x + 6 = 100 - 6 - 6

Get the term with the exponent alone

10-12x = 94log1094 = -12x

Use your calculator to get the answer log(94)/log(10)

Change to log form

1.973 = -12x-12 -12- .164 = x

Solving equations with e:4e2x = 5 Get the term with the

exponent alone

e2x = 1.25loge1.25 = 2x

Use your calculator to get the answer ln(1.25)

Change to log form

.2231 = 2x

4 4

Change to ln form

ln1.25 = 2x

2 2.1116 = x

One More Example:4 – 2ex = -23 Get the term with the

exponent alone

-2ex = -27

Use your calculator to get the answer ln(13.5)

Change to log form

loge(13.5) = x Change to ln form

ex = 13.5

-4 - 4

-2 -2

ln(13.5) = x2.6 = x