Answer :
Certainly! Let's simplify the expression step-by-step: [tex]\[\log_9 10 - \log_9 \frac{1}{2} - \log_9 4\][/tex]
1. Combine the first two logarithms:
[tex]\[\log_9 10 - \log_9 \frac{1}{2}\][/tex]
Using the logarithmic property [tex]\(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)[/tex]:
[tex]\[\log_9 \left(\frac{10}{\frac{1}{2}}\right)\][/tex]
2. Simplify the fraction inside the logarithm:
[tex]\[ \frac{10}{\frac{1}{2}} = 10 \times 2 = 20 \][/tex]
So, the expression becomes:
[tex]\[ \log_9 20 \][/tex]
3. Combine [tex]\(\log_9 20\)[/tex] with [tex]\(\log_9 4\)[/tex]:
[tex]\[ \log_9 20 - \log_9 4 \][/tex]
Again, using the logarithmic property [tex]\(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)[/tex]:
[tex]\[ \log_9 \left(\frac{20}{4}\right) \][/tex]
4. Simplify the fraction inside the logarithm:
[tex]\[ \frac{20}{4} = 5 \][/tex]
So the expression simplifies to:
[tex]\[ \log_9 5 \][/tex]
Therefore, the expression [tex]\[\log_9 10 - \log_9 \frac{1}{2} - \log_9 4 \][/tex] simplifies to [tex]\(\log_9 5\)[/tex].
1. Combine the first two logarithms:
[tex]\[\log_9 10 - \log_9 \frac{1}{2}\][/tex]
Using the logarithmic property [tex]\(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)[/tex]:
[tex]\[\log_9 \left(\frac{10}{\frac{1}{2}}\right)\][/tex]
2. Simplify the fraction inside the logarithm:
[tex]\[ \frac{10}{\frac{1}{2}} = 10 \times 2 = 20 \][/tex]
So, the expression becomes:
[tex]\[ \log_9 20 \][/tex]
3. Combine [tex]\(\log_9 20\)[/tex] with [tex]\(\log_9 4\)[/tex]:
[tex]\[ \log_9 20 - \log_9 4 \][/tex]
Again, using the logarithmic property [tex]\(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)[/tex]:
[tex]\[ \log_9 \left(\frac{20}{4}\right) \][/tex]
4. Simplify the fraction inside the logarithm:
[tex]\[ \frac{20}{4} = 5 \][/tex]
So the expression simplifies to:
[tex]\[ \log_9 5 \][/tex]
Therefore, the expression [tex]\[\log_9 10 - \log_9 \frac{1}{2} - \log_9 4 \][/tex] simplifies to [tex]\(\log_9 5\)[/tex].
