This is an essential skill to be learned in this chapter. We can use the properties of the logarithm to combine expressions involving logarithms into a single logarithm with coefficient 1.A logarithmic expression is completely expanded when the properties of the logarithm can no further be applied. Solution for Condensing a Logarithmic Expression In Exercises 61-76, condense the expression to the logarithm of a single quantity. Here is an alternate proof of the quotient rule for. We can use the properties of the logarithm to expand logarithmic expressions using sums, differences, and coefficients. We can use the power rule to expand logarithmic expressions involving negative and fractional exponents.Since the natural logarithm is a base- e logarithm, ln x = log e x, all of the properties of the logarithm apply to it.The power property of the logarithm allows us to write exponents as coefficients: log b x n = n log b x. ![]() Rule 3: Power Rule The logarithm of an exponential number is the exponent times the logarithm of the base. The quotient property of the logarithm allows us to write a quotient as a difference: log b ( x y ) = log b x − log b y. The logarithm of the quotient of numbers is the difference of the logarithm of individual numbers.Since we have condensed or compressed three logarithmic. That means we can convert those addition operations (plus symbols) outside into multiplication inside. The product property of the logarithm allows us to write a product as a sum: log b ( x y ) = log b x + log b y. Example 1: Combine or condense the following log expressions into a single logarithm: This is the Product Rule in reverse because they are the sum of log expressions.A fourth root is the same as the one-fourth power Condense the logarithms using the product and quotient rule. ![]() A fourth root is the same as the one-fourth power: Condense the logarithms using the product and quotient rule. A square root is the same as the one-half power. A square root is the same as the one-half power. The coefficient of 1/6 on the middle term becomes the power on the expression inside the logarithm: A radical can be written as a fractional power. The inverse properties of the logarithm are log b b x = x and b log b x = x where x > 0. The coefficient of 1/6 on the middle term becomes the power on the expression inside the logarithm A radical can be written as a fractional power.Given any base b > 0 and b ≠ 1, we can say that log b 1 = 0, log b b = 1, log 1 / b b = − 1 and that log b ( 1 b ) = − 1.When condensing logarithms, our goal is to compress the expressions altogether by using different logarithmic properties. The next section will show you how condensing logarithms is the opposite of expanding logarithms. Using exponent properties, this is log3 38 and. ![]()
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