Find the logarithm using the change of base formula. [tex]$\log _6 15$[/tex]
Answer (1)
Apply the change of base formula: lo g b a = l o g c b l o g c a .
Rewrite the expression: lo g 6 15 = l o g 10 6 l o g 10 15 or lo g 6 15 = l n 6 l n 15 .
Calculate the value: l o g 10 6 l o g 10 15 ≈ 1.5114 .
The final answer is: 1.5114 .
Explanation
Understanding the Problem We are asked to find the value of lo g 6 15 using the change of base formula. The change of base formula allows us to rewrite a logarithm in terms of logarithms with a different base.
Change of Base Formula The change of base formula states that lo g b a = l o g c b l o g c a for any positive a , b , and c where b = 1 and c = 1 . We can use this formula to rewrite lo g 6 15 using a common base, such as base 10 or the natural logarithm (base e ).
Applying the Formula Using base 10, we have lo g 6 15 = l o g 10 6 l o g 10 15 . Alternatively, using the natural logarithm, we have lo g 6 15 = l n 6 l n 15 .
Calculating the Value Using a calculator, we find that l o g 10 6 l o g 10 15 ≈ 1.5114 and l n 6 l n 15 ≈ 1.5114 .
Final Answer Therefore, lo g 6 15 ≈ 1.5114 .
Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. The change of base formula is particularly useful when a calculator only has logarithms for certain bases (like base 10 or base e) but you need to calculate a logarithm with a different base. For example, if you want to compare the relative intensity of two earthquakes measured on different scales, you might need to convert logarithms from one base to another.
