If log_75(6) = a log_105(36) = b log_6(7/5) = c then relation between a, b and c is:

We are given the logarithmic equations:

\[
\log_{75}(6) = a, \quad \log_{105}(36) = b, \quad \log_{6}\left(\frac{7}{5}\right) = c
\]

We need to find the relation between \(a\), \(b\), and \(c\).

### Step 1: Express the logarithms in terms of natural logs (or base 10 logs)
Using the change of base formula, we convert all the logarithms to a common base. The change of base formula is:

\[
\log_b(x) = \frac{\log(x)}{\log(b)}
\]

Thus, we can rewrite each logarithmic expression as follows:

1. \(\log_{75}(6) = a\):

\[
a = \frac{\log(6)}{\log(75)}
\]

2. \(\log_{105}(36) = b\):

\[
b = \frac{\log(36)}{\log(105)}
\]

3. \(\log_6\left(\frac{7}{5}\right) = c\):

\[
c = \frac{\log\left(\frac{7}{5}\right)}{\log(6)} = \frac{\log(7) - \log(5)}{\log(6)}
\]

### Step 2: Find relations between \(a\), \(b\), and \(c\)

From the expressions of \(a\), \(b\), and \(c\), we notice that all terms involve the logarithms of various numbers. To simplify the relation, we will manipulate these expressions.

Start by isolating \(\log(6)\) from \(a\)'s equation:

\[
a = \frac{\log(6)}{\log(75)} \quad \Rightarrow \quad \log(6) = a \cdot \log(75)
\]

Substitute \(\log(6)\) in \(c\)'s expression:

\[
c = \frac{\log(7) - \log(5)}{a \cdot \log(75)}
\]

For \(b\), express \(\log(36)\) as \(2\log(6)\):

\[
b = \frac{\log(36)}{\log(105)} = \frac{2\log(6)}{\log(105)}
\]

Now substitute \(\log(6)\) from the expression of \(a\):

\[
b = \frac{2a \cdot \log(75)}{\log(105)}
\]

### Final Relation:
Thus, we now have expressions for \(b\) and \(c\) in terms of \(a\), \(\log(75)\), and \(\log(105)\). The precise relationship depends on the numerical values of the logs, but we now see that \(b\) is proportional to \(a\), and \(c\) is inversely related to \(a\), involving \(\log(75)\), \(\log(105)\), \(\log(7)\), and \(\log(5)\).

Would you like further details on simplifying this relationship, or solving numerically?

If log_75(6) = a, log_105(36) = b, log_6(7/5) = c then relation between a, b and c is:

Explore the relationship between a, b, and c, given the logarithmic expressions log_75(6) = a, log_105(36) = b, and log_6(7/5) = c. This solution utilizes the change of base formula and other logarithmic properties to establish a connection between these variables. Suitable for students at an advanced high school or college level.

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