Find the logarithm using the change of base formula. [tex]$\log _8 18$[/tex] [tex]$\log _8 18=$ $\square$[/tex] (Round to four decimal places as needed.)
Answer (2)
Apply the change of base formula: lo g 8 18 = l o g 10 8 l o g 10 18 .
Calculate the values of the logarithms: lo g 10 18 ≈ 1.2553 and lo g 10 8 ≈ 0.9031 .
Divide the logarithms: 0.9031 1.2553 ≈ 1.3900 .
Round the result to four decimal places: 1.3900 .
Explanation
Understanding the problem We are asked to find the value of lo g 8 18 using the change of base formula and round the result to four decimal places. The change of base formula allows us to convert a logarithm from one base to another.
Applying the Change of Base Formula The change of base formula is given by: lo g b a = lo g c b lo g c a where a , b , and c are positive numbers and b = 1 and c = 1 . We can use either the common logarithm (base 10) or the natural logarithm (base e ) for the new base. Here, we will use the common logarithm (base 10).
Setting up the Calculation Using the change of base formula, we have: lo g 8 18 = lo g 10 8 lo g 10 18
Calculating the Logarithms We know that lo g 10 18 ≈ 1.2553 and lo g 10 8 ≈ 0.9031 . Therefore, lo g 8 18 = 0.9030899869919435 1.255272505103306 ≈ 1.3899750004807707
Rounding to Four Decimal Places Rounding the result to four decimal places, we get: lo g 8 18 ≈ 1.3900
Final Answer Therefore, the logarithm of 18 with base 8, rounded to four decimal places, is approximately 1.3900.
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes (the Richter scale), the loudness of sounds (decibels), and the acidity of a solution (pH). The change of base formula is particularly useful when you need to calculate a logarithm on a calculator that only has buttons for common logarithms (base 10) or natural logarithms (base e). For example, if you want to find the logarithm of a number with base 2, you can use the change of base formula to convert it to a base 10 or base e logarithm, which you can then calculate using a calculator.
Using the change of base formula, lo g 8 18 can be calculated as l o g 10 8 l o g 10 18 . This results in a value of approximately 1.3900 when rounded to four decimal places.
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