Question

I have a practice problem that I need help on Make sure to answer the following:Where did Arjun make his errors?Explain his errors AND the properties of logarithms that leads to the correct answer State the correct answer.

Answer 1

Arjun added the logarithm arguments instead of using the properties of the logarithm function to correctly add the three logarithms.

When we add two logarithms that have the same basis, this is equivalent to finding the logarithm of the product of the arguments:

$\log _ab+\log _ac+\log _ad=_{}\log _a(b\cdot c\cdot d)$

Also, we need to use the following property:

$-\log _aA=\log _aA^{-1}=\log _a\frac{1}{A}$

So, for the given expression, the correct solution is:

$\log _7x+\log _7y-\log _7z=\log _7(x\cdot y)+\log _7\frac{1}{z}=\log _7(x\cdot y\cdot\frac{1}{z})$

This result can also be written as:

$\log _7(\frac{xy}{z})$