Simplify:
1. 2log10^5+log10^4
2. 2log3^6-log3^4
3. log4^40-log4^5
4. log4^3-log4^48
I need to see all of the steps.
Thanks
You need to review the rules:
log a^n= n log a
log4^40 -log4^5
40log4-5log3=35log4
all of these are the same skill.
To simplify logarithmic expressions, we can use the following logarithmic identities:
1. log a + log b = log(ab)
2. log a - log b = log(a/b)
3. log a^b = b log a
Now let's simplify each expression step by step:
1. 2log10^5 + log10^4
Using the first identity, we can rewrite the expression as:
log10^5^2 + log10^4
Now, we can apply the third identity to simplify further:
log10^(5^2) + log10^4
= log10^25 + log10^4
Applying the first identity again:
log10^(25 * 4)
= log10^100
So the simplified expression is log10^100.
2. 2log3^6 - log3^4
Using the third identity, the expression becomes:
log3^(6^2) - log3^4
= log3^36 - log3^4
Using the first identity, we have:
log3^(36 / 4)
= log3^9
So the simplified expression is log3^9.
3. log4^40 - log4^5
Using the third identity, we can rewrite the expression as:
40 log4 - 5 log4
Using the second identity, we have:
log4^(40 / 5)
= log4^8
So the simplified expression is log4^8.
4. log4^3 - log4^48
Using the third identity, the expression becomes:
3 log4 - 48 log4
Using the second identity, we have:
log4^(3 / 48)
= log4^(1/16)
So the simplified expression is log4^(1/16).
I hope this helps! Let me know if you have any further questions.