Introduction Previously, you learned how to graph logarithmic equations with bases other than 10. It may be necessary to convert other bases to common logarithms if you are using a calculator that does not automatically do this. This process involves applying a property of logarithms in order to convert logarithms with different bases to common logarithms. Properties of logarithms allow you to solve problems containing logarithms by making the logarithms easier to work with. Applying logarithmic properties often allows for simpler and fewer calculations in a problem : Properties of Logarithms

Key Concepts Recall that a logarithm is a quantity that represents the power to which a base b must be raised in order to equal a quantity x. It is typically written in the form log b x. Recall that when logarithms have bases other than 10, you can rewrite them as common logarithms by using the change of base formula, : Properties of Logarithms

Key Concepts, continued Since logarithms are the inverses of exponents, they have similar properties. Like exponents, logarithms can be added, subtracted, multiplied, and divided. Performing operations with logarithms is the same as performing operations on exponents. For example, log (ab) = log a + log b is the same as multiplying two numbers raised to powers, as in x a x b = x a + b : Properties of Logarithms

Key Concepts, continued The following table includes properties of logarithms and corresponding properties of exponents : Properties of Logarithms Exponent propertyLogarithm property Logarithm property name a x a y = a x + y log a (x y) = log a x + log a y Product Property of Logarithms Quotient Property of Logarithms (a x ) y = a x y log a x y = y log a x Power Property of Logarithms

Common Errors/Misconceptions misunderstanding the parts of a logarithmic equation misunderstanding how to convert a logarithmic equation to an exponential equation or vice versa making calculator input errors : Properties of Logarithms

Guided Practice Example 2 Use the Product Property of Logarithms to show that log a (xy) = log a x + log a y. Verify this property mathematically : Properties of Logarithms

Guided Practice: Example 2, continued 1.Write the equivalent exponential equation for log a x. To write the equivalent exponential equation, let M = log a x. By the definition of logarithms, M = log a x is equivalent to a x = M : Properties of Logarithms

Guided Practice: Example 2, continued 2.Write the equivalent exponential equation for log a y. To write the equivalent exponential equation, let N = log a y. By the definition of logarithms, N = log a y is equivalent to a y = N : Properties of Logarithms

Guided Practice: Example 2, continued 3.Write the exponential form of the Product Property. According to the Product Property, log a (xy) = M + N. Therefore, the exponential form of the Product Property is a (M + N) = xy. By the Product Property of Exponents, a (M + N) = (a M )(a N ) : Properties of Logarithms

Guided Practice: Example 2, continued 4.Substitute specific values for a, x, and y into the original equation to verify that the Product Property of Logarithms holds true for this problem. The original equation was log a (xy) = log a x + log a y. Start by substituting values into the left-hand side of the equation, log a (xy), and evaluating. Then, substitute the same values into log a x + log a y and solve to see if the result matches that for log a (xy) : Properties of Logarithms

Guided Practice: Example 2, continued For example, let a = 10, x = 3, and y = 5. log a (xy)Original expression = log (10) [(3)(5)]Substitute 10 for a, 3 for x, and 5 for y. = log 10 15Multiply the numbers. ≈ 1.18Use a calculator to compute log : Properties of Logarithms

Guided Practice: Example 2, continued Now evaluate the terms in the right-hand side of the equation, log a x + log a y, for the same values of a, x, and y. log a x + log a yOriginal expression = log (10) (3) + log (10) (5)Substitute 10 for a, 3 for x, and 5 for y. ≈ Use a calculator to compute log 10 3 and log ≈ 1.18Add : Properties of Logarithms

Guided Practice: Example 2, continued The results match: log ≈ 1.18 and log log 10 5 ≈ Therefore, the Product Property of Logarithms holds true : Properties of Logarithms ✔

Guided Practice: Example 2, continued : Properties of Logarithms

Guided Practice Example 3 Use the properties of logarithms to simplify the logarithmic function f(x) = 2 log x – log (x – 3) : Properties of Logarithms

Guided Practice: Example 3, continued 1.Identify which logarithmic properties can be used to simplify the logarithmic function. The given function contains the term 2 log x. This is the same as 2 log x. This form is also seen in the Power Property of Logarithms, where log a x y = y log a x : Properties of Logarithms

Guided Practice: Example 3, continued Additionally, this function includes the subtraction of two logarithmic terms. This form is also seen in the Quotient Property of Logarithms, where Both the Power Property of Logarithms and the Quotient Property of Logarithms can be used to simplify the given function : Properties of Logarithms

Guided Practice: Example 3, continued 2.Simplify the logarithmic function by applying the identified properties. Apply the Power Property of Logarithms to the term 2 log x from the original function. The base, a, is understood to be 10 and is omitted. y log x = log x y Power Property of Logarithms (2) log x = log x (2) Substitute 2 for y. Thus, 2 log x = log x : Properties of Logarithms

Guided Practice: Example 3, continued Now apply the Quotient Property, to the original function. Use log x 2 in place of 2 log x. f(x) = 2 log x – log (x – 3)Original function f(x) = (log x 2 ) – log (x – 3) Substitute log x 2 for 2 log x. Simplify using the Quotient Property of Logarithms : Properties of Logarithms

Guided Practice: Example 3, continued The logarithmic function f(x) = 2 log x – log (x – 3) simplifies to : Properties of Logarithms ✔

Guided Practice: Example 3, continued : Properties of Logarithms