SOLUTION: Log 6 to the base 2+ log 7 to the base 2 equals to log a to the base 2 Solve for a

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Question 1205462: Log 6 to the base 2+ log 7 to the base 2 equals to log a to the base 2
Solve for a
Found 2 solutions by Theo, math_tutor2020:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
log2(6) + log2(7) = log2(6*7) = log2(42)

you can use your calculator to confirm by using the log base conversion formula of log2(x) = log10(x)/log(2).

the log function of your calculator is the same as log10.
using your calculator log funciton, then log2(x) = log(x)/log(2)

your problem want you to find A in the equation of log2(6) + log2(7) = log2(A).

one of the log rules states that logb(a) + logb(b) = logb(a*b).
the base can be any valid log base.

consequently log2(6) + log2(7) = log2(6*7) = log2(42).

A = 42 is your answer.

you can confirm in your calculator by using the lob base conversion formula to get:

log2(6) + log2(7) = log2(42)

in your calculator, log2(6) = log(6)/log(2) and log2(7) = log(7)/log(2)
and log2(42) = log(42)/log(2).

you get log(6)/log(2) + log(7)/log(2) = (log(6) + log(7))/log(2) = 5.392317423 and you get log(42)/log(2) = the same.

this confirms log2(6) + log2(7) is equivalent to log2(42).

here's some references for you that might help clarify your understanding of the log rules.

log change of base formula:
https://www.chilimath.com/lessons/advanced-algebra/change-of-base-formula-or-rule/

log rules:
https://www.chilimath.com/lessons/advanced-algebra/logarithm-rules/


Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Useful log rules that we'll need:
  1. log(M)+log(N) = log(M*N)
  2. If log(M) = log(N), it must mean M = N.
The second rule is valid because the log function of any base is one-to-one aka injective.

log%282%2C%286%29%29%2Blog%282%2C%287%29%29=log%282%2C%28a%29%29

log%282%2C%286%2A7%29%29=log%282%2C%28a%29%29 Use the lst rule mentioned above

log%282%2C%2842%29%29=log%282%2C%28a%29%29

42+=+a Use the 2nd rule mentioned

a+=+42

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Often it's a pain to repeatedly write "base 2" whether in word form or symbolically.
To avoid this repetition and speed things up, you can state somewhere at the top of your page "all logs mentioned are base 2" or something along those lines.

After that statement you could have your starting equation as:
log(6) + log(7) = log(a)
Just keep in mind that these are base 2 logs of course.

It turns out the base doesn't matter here. The log rules mentioned at the top work for any base as long as it's a positive real number and the base isn't 1. The final answer a = 42 will happen for any valid base.

Bonus exercise: Explain to your classmate why log base 1 isn't valid. A hint is to think about the change of base formula. Another hint is to look at the denominator.