How do we use properties to simplify logarithmic expressions? Properties of Logarithms Essential Questions How do we use properties to simplify logarithmic expressions? How do we translate between logarithms in any base? Holt McDougal Algebra 2 Holt Algebra 2

The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH = [H+]. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

Remember that to multiply powers with the same base, you add exponents.

The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified. Helpful Hint Think: log j + log a + log m = log jam

log64 + log69 Express as a single logarithm. Simplify, if possible. Adding Logarithms Express as a single logarithm. Simplify, if possible. log64 + log69 To add the logarithms, multiply the numbers. log6 (4  9) log6 36 = x Simplify. 2 Think: 6 x = 36.

Adding Logarithms Express as a single logarithm. Simplify, if possible. log5625 + log525 To add the logarithms, multiply the numbers. log5 (625 • 25) log5 15,625 Simplify. = x 6 Think: 5 x = 15625.

Express as a single logarithm. Simplify, if possible. Adding Logarithms Express as a single logarithm. Simplify, if possible. To add the logarithms, multiply the numbers. = x Simplify. Think: x = 3 1 3 –1

Remember that to divide powers with the same base, you subtract exponents Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

The property above can also be used in reverse. Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Caution

Subtracting Logarithms Express as a single logarithm. Simplify, if possible. To subtract the logarithms, divide the numbers. = x Simplify. 2 Think: 5 x = 25.

Subtracting Logarithms Express as a single logarithm. Simplify, if possible. To subtract the logarithms, divide the numbers. = x Simplify. 1 Think: 7 x = 7.

Because you can multiply logarithms, you can also take powers of logarithms.

Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. Powers expressed as multiplication. Think: 2 x = 32. 30 Simplify.

Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. Powers expressed as multiplication. Think: 16 x = 4. 10 Simplify.

Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. Powers expressed as multiplication. Think: 10 x = 10. 4 Simplify.

Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. Powers expressed as multiplication. Think: 5 x = 25. 4 Simplify.

Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. Powers expressed as multiplication. Think: 2 x = ½ . -5 Simplify.

Exponential and logarithmic operations undo each other since they are inverse operations.

Recognizing Inverses Simplify each expression. 12. log381 13. 5log510 11. log3311 log3311 log334 5log510 11 4 10 14. log100.9 15. 2log2(8x) Log10100.9 2log2(8x) 0.9 8x

Most calculators calculate logarithms only in base 10 or base e Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

Changing the Base of a Logarithm 16. Evaluate log328. 17. Evaluate log927. Change to base 10 Change to base 10 log328 = log8 log32 log927 = log27 log9 ≈ 0.6 ≈ 1.5 18. Evaluate log816. Change to base 10 log816 = log16 log8 ≈ 1.3

Lesson 9.1 Practice A