Math Problem Statement

Let \log_{b}\left(2\right)=p $$$$ \log_{b}\left(3\right)=q $$$$ \log_{b}\left(5\right)=r. Express log5(b)\log_5\left(b\right)in terms of p, q, or r

Solution

To express log5(b)\log_5(b) in terms of pp, qq, and rr, we can use the change of base formula, which states:

loga(c)=logb(c)logb(a)\log_a(c) = \frac{\log_b(c)}{\log_b(a)}

Here, we want to express log5(b)\log_5(b). Using the change of base formula:

log5(b)=logb(b)logb(5)\log_5(b) = \frac{\log_b(b)}{\log_b(5)}

We know that logb(b)=1\log_b(b) = 1 and logb(5)=r\log_b(5) = r, so we have:

log5(b)=1r\log_5(b) = \frac{1}{r}

Thus, the expression for log5(b)\log_5(b) in terms of rr is:

log5(b)=1r\log_5(b) = \frac{1}{r}

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

  1. How can log5(2)\log_5(2) be expressed in terms of pp and rr?
  2. What is the relationship between logb(30)\log_b(30) and pp, qq, and rr?
  3. How would you express log3(b)\log_3(b) in terms of pp, qq, and rr?
  4. Can you express log2(3)\log_2(3) in terms of pp and qq?
  5. How do you find logb2(2)\log_{b^2}(2) using pp and the change of base formula?

Tip: When solving logarithmic problems, the change of base formula is a powerful tool for converting logs to different bases, allowing for easier manipulation and comparison of logarithmic expressions.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula

Formulas

Change of base formula: \(\log_a(c) = \frac{\log_b(c)}{\log_b(a)}\)

Theorems

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Suitable Grade Level

High School