We will start by applying this law to the first two terms, giving us Next, we can simplify the subtractions by using the division law L o g l o g l o g l o g l o g l o g 1 2 8 0 − 2 2 − 5 = 1 2 8 0 − 4 − 5. Substituting this back into the original expression, we then get ![]() To simplify 2 2 l o g , we use the power law of We will start by simplifying the term with the multiplication since this is a priority in the order of operations. We have two operations in this expression, a multiplication and The laws of logarithms with the same base, since both terms have the sameīase. To calculate l o g l o g l o g 1 2 8 0 − 2 2 − 5, we can use Having derived the three logarithm laws for like bases, we summarize them in the following properties.Įxample 2: Computing Logarithmic Expressions Using Laws of Logarithmsįind the value of l o g l o g l o g 1 2 8 0 − 2 2 − 5 without using a calculator. Then, substituting □ = □ l o g in the logarithmic If we rewrite this as a logarithmic equation If we then raise both sides to the power of □Īnd use the power law for exponents, we get Let’s say we have the logarithmic equation □ = □ l o g that has base □, where □ > 0, and argument □ > 0. Lastly, we will derive the power law of logarithms, which links to the power law for exponents. By rewriting these as exponents, we getĭividing the first exponential equation by the second and using the Which have base □, where □ > 0, and arguments □, □ > 0. ![]() Let’s say we have two logarithmic equations Next, we will derive the division law of logarithms that links to the division law for exponents. ![]() Then, substituting □ = □ l o g and □ = □ l o g , we get If we rewrite this as a logarithmic equation with base Multiplying these two exponential equations together and using the multiplication law for exponents, Let’s say we have two logarithmic equations □ = □ l o g and □ = □ l o g , both of which have base □, where □ > 0, and arguments □, □ > 0. We will now derive each of the laws of logarithms for like bases in turn.įirst, we will derive the multiplication law of logarithms, which links to the multiplication law for exponents. Property: Composition of Exponential and Logarithmic FunctionsĪlso, note that □ > 0 is a requirement for the first property since we cannot take the logarithm of a negative number.
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