Change all log to base 10 (lg). Use the change of base formula.

$latex \displaystyle \frac{{\lg 7}}{{\lg 2}}\times \frac{{\lg 8}}{{\lg 3}}\div \frac{{\lg \sqrt{7}}}{{\lg 9}}$

$latex \displaystyle \frac{{\lg 7}}{{\lg 2}}\times \frac{{\lg {{2}^{3}}}}{{\lg 3}}\div \frac{{\lg {{7}^{{\frac{1}{2}}}}}}{{\lg {{3}^{2}}}}$

Use the formula log_{a}b^{c}= c log_{a}b. Simplify the expression.

$latex \displaystyle \frac{{\lg 7}}{{\lg 2}}\times \frac{{3\lg 2}}{{\lg 3}}\div \frac{{\frac{1}{2}\lg 7}}{{2\lg 3}}$

$latex \displaystyle\frac{{3\lg 7}}{{\lg 3}}\div \frac{{\frac{1}{2}\lg 7}}{{2\lg 3}}$

$latex \displaystyle\frac{{3\lg 7}}{{\lg 3}}\times \frac{{2\lg 3}}{{\frac{1}{2}\lg 7}}$

$latex \displaystyle\frac{3}{1}\times \frac{2}{{\frac{1}{2}}}$

12 (Shown)

Normally student will change the expression to the same base number. But it will be difficuil for the bove question. A better solution is to change all to base 10 (lg), the cancel out the common logs. You may also change to ln (log_{e}) instead of lg., the answer will still be the same.

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