Devonte used the change of base formula to approximate [tex]$\log _8 25$[/tex]. Which expression did Devonte use?
A. [tex]$\log \frac{8}{25}$[/tex]
B. [tex]$\frac{\log 8}{\log 25}$[/tex]
C. [tex]$\log \frac{25}{8}$[/tex]
D. [tex]$\frac{\log 25}{\log 8}$[/tex]
Answer (1)
The problem requires applying the change of base formula to lo g 8 25 .
The change of base formula is lo g a b = l o g c a l o g c b .
Applying the formula, lo g 8 25 = l o g 8 l o g 25 .
The expression Devonte used is lo g 8 lo g 25 .
Explanation
Understanding the Problem Devonte used the change of base formula to approximate lo g 8 25 . We need to determine which expression Devonte used. The change of base formula states that lo g a b = l o g c a l o g c b for any valid base c .
Applying Change of Base Formula Applying the change of base formula to lo g 8 25 , we can use a common base, such as base 10. This gives us lo g 8 25 = l o g 10 8 l o g 10 25 .
Simplifying the Expression Since the base is not specified in the options, we assume 'log' refers to the common logarithm (base 10). Therefore, lo g 8 25 = l o g 8 l o g 25 .
Identifying the Correct Expression Comparing this result with the given options, we find that the correct expression is l o g 8 l o g 25 .
Final Answer Therefore, the expression Devonte used is lo g 8 lo g 25 .
Examples
The change of base formula is useful in many real-world applications, especially when dealing with logarithmic scales. For example, in seismology, the Richter scale uses logarithms to measure the magnitude of earthquakes. If you need to compare the energy released by an earthquake measured on one base to another, the change of base formula becomes essential. Similarly, in finance, when comparing interest rates or investment growth rates calculated using different compounding periods, the change of base formula helps standardize the comparison.
