11.3 – Exponential and Logarithmic Equations

CHANGE OF BASE FORMULA

6826.15log

15log15log5

b

aab log

loglog

Ex: Rewrite log515 using the change of base formula

Steps for solving exponential equations

Take a common logarithm of each side

Use the power property of logarithms

Solve for x by dividing Use a calculator to find the

approximate value

Review exponential equations

64 = 2 = 2

Solving Exponential Equations

43 x

43 x

4log3log x

3log

4logx

4log3log x 1. Take the log of both sides

2. Use the power property

3. Solve for x.

Solve . Round to the nearest ten-thousandth.

X=1.2619 4. Use a calculator.

Check your answer – 31.2619=4

Another Example

1013 4 x

1013 4 x

101log3log)4( x

3log

101log4 x

101log3log 4 x 1. Take the log of both sides

2. Use the power property

3. Solve for x.

Solve . Round to the nearest ten-thousandth.

X=4.2009 – 4 = 0.2009 4. Use a calculator.

Check your answer – 30.2009+4=101

Let’s try some

1505x 802 x4

Let’s try some

1505x 802 x4

Let’s try some

735 x 2073 x 1003 4 x

Let’s try some

735 x 2073 x 1003 4 x

CHANGE OF BASE – HOW IT WORKS

Use the change of base formula to evaluate . Then convert it to a logarithm of base 2.

15log3

4650.215log3

x23 log15log

3log

15log15log3 1. Rewrite using the

change of base formula

2. Use a calculator

3. Write an equation to convert to base 2

CHANGE OF BASE – HOW IT WORKS

x2log4650.2

2log

log4650.2

x

xlog7420.0

7420.010x

xlog2log 4650.2 6. Multiply both sides of the equation by log2

7. Use a calculator; simplify.

8. Write in exponential form.

5. Rewrite using the change of base formula

4. Substitute log315=2.4650

X=5.5208 9. Use a calculator.

Log315 is approximately equal to 2.4650 or log25.5208

Let’s try one

Use the change of base formula to evaluate . Then convert it to a logarithm of base 8.

400log5

7227.3400log5

x85 log400log

5log

400log400log5 1. Rewrite using the

change of base formula

2. Use a calculator

3. Write an equation to convert to base 2

x8log7227.3

8log

log7227.3

x

xlog3619.3

3619.310x

xlog8log .72273 6. Multiply both sides of the equation by log8

7. Use a calculator; simplify.

8. Write in exponential form.

5. Rewrite using the change of base formula

4. Substitute log5400=3.727

X=2301 9. Use a calculator.

Log5400 is approximately equal to 3.7227 or log82301

SOLVING SIMPLE LOG EQUATIONS

x642

3

8

6

16x

26log

232log

4

4

x

x

2)3(log2log solve tologs of properties Use 44 x

1. Use the product property

2. Write in exponential form.

x616

2)3(logx2log 44

3. Simplify

4. Solve for x.

Let’s try some

1643 555 loglogxlog

Let’s try some

1643 555 loglogxlog

Let’s try some

64log4logxlog2

Let’s try some

64log4logxlog2

Solving exponential equations with a graphing calculator

150062 x

1. Type two equations into y=

Solution: 2.0408

2. Graph. Suggest Zoom fit (0)especially for large values

3. Use the calc function to find the intersection of the two graphs.