When the concentration of hydrogen ions is doubled, the decreases by about. If the concentration is doubled, the new concentration is. Suppose is the original concentration of hydrogen ions, and is the original of the liquid. If the concentration of hydrogen ions in a liquid is doubled, what is the effect on ? Next we rearrange and apply the product rule to the sum: Next we apply the product rule to the sum:įinally, we apply the quotient rule to the difference:ĮXAMPLE 11 Rewriting as a Single Logarithm Rewrite differences of logarithms as the logarithm of a quotient.ĮXAMPLE 9 Using the Product and Quotient Rules to Combine LogarithmsĮXAMPLE 10 Condensing Complex Logarithmic Expressions Rewrite sums of logarithms as the logarithm of a product. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm. We will learn later how to change the base of any logarithm before condensing. It is important to remember that the logarithms must have the same base to be combined. We can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. We can expand by applying the Product and Quotient Rules. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.ĮXAMPLE 8 Expanding Complex Logarithmic Expressions Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression Then seeing the product in the first term, we use the product rule:įinally, we use the power rule on the first term: Remember, however, that we can only do this with products, quotients, powers, and roots – never with addition or subtraction inside the argument of the logarithm.ĮXAMPLE 6 Expanding Logarithms Using Product, Quotient, and Power Rulesįirst, because we have a quotient of two expressions, we can use the quotient rule: With practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. We can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power: We can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Sometimes we apply more than one rule in order to simplify an expression. Taken together, the product rule, quotient rule, and power rule are often called "laws of logs".
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