Express as a logarithm of a single number or expression:
1. 5log4^p+log4^q
2. log10^x-4 log10^y
3. 4log3^A-1/2 log3^B
4. log5^M+1/4 log5^N
5. log2^M+log2^N+3
6. log5^x-log5^y+2
7. 1-3 log5^x
8. (1+log9^x)/2
I need to see all of the steps.
Thanks
Sure! Let's solve each expression step by step:
1. To express 5log4^p + log4^q as a logarithm of a single number or expression, we use the logarithmic property log(ab) = log(a) + log(b).
Applying this property, we can rewrite the expression as:
log4^(4^p) * log4^q
2. To express log10^x - 4log10^y as a logarithm of a single number or expression, we can use the logarithmic property log(a/b) = log(a) - log(b).
Applying this property, we get:
log10^x - log10^(y^4)
3. To express 4log3^A - (1/2)log3^B as a logarithm of a single number or expression, we can rearrange it using logarithmic properties.
We can rewrite the expression as:
log3^(3^A)^4 - log3^(B^(1/2))
4. To express log5^M + (1/4)log5^N as a logarithm of a single number or expression, we can rewrite it using logarithmic properties.
The expression becomes:
log5^(M * 5^(N^(1/4)))
5. To express log2^M + log2^N + 3 as a logarithm of a single number or expression, we use the logarithmic property log(ab) = log(a) + log(b).
Applying this property, we get:
log2^(2^M) + log2^(2^N) + log2^2^2
6. To express log5^x - log5^y + 2 as a logarithm of a single number or expression, we can rearrange it using logarithmic properties.
The expression becomes:
log5^(x/y) + log5^25
7. To express 1 - 3log5^x as a logarithm of a single number or expression, we can rewrite it using logarithmic properties.
The expression becomes:
log5^5 - log5^x^3
8. To express (1 + log9^x) / 2 as a logarithm of a single number or expression, we can use logarithmic properties to rearrange it.
The expression becomes:
log9^√x * log9^9
Please note that in these step-by-step explanations, we have applied known logarithmic properties such as log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), and log(a^b) = b * log(a). These properties can help simplify logarithmic expressions and express them as a logarithm of a single number or expression.