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## Algebra (all content)

### Course: Algebra (all content)>Unit 11

Lesson 25: The change of base formula for logarithms (Algebra 2 level)

# Evaluating logarithms: change of base rule

Sal approximates log₅(100) by rewriting it as log(100)/log(5) using the change of base rule, then evaluates with a calculator. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• I posted an answer to this question with an alternate proof to the change of base formula that I find more intuitive. I did this because neither the comment nor question sections have enough space for it.
• Let logₐ(b) = s. Then we wish to show that, for any value of x, logₐ(b) = logₓ(b) / logₓ(a). Let's say logₓ(a) = t and logₓ(b) = u.

Change each of these to the exponent notation:

logₐ(b) = s means aˢ = b.

logₓ(a) = t means xᵗ = a.

logₓ(b) = u means xᵘ = b.

So we know that aˢ = xᵘ, because they both equal b. We also know a = xᵗ now, so we can substitute xᵗ for a in the equation aˢ = xᵘ, and get (xᵗ)ˢ = xᵘ. This simplifies to xˢᵗ = xᵘ using the exponent properties, which implies st = u, or s = u/t. Recall that logₐ(b) = s, logₓ(a) = t, and logₓ(b) = u. So we've now shown with s = u/t that logₐ(b) = logₓ(b) / logₓ(a). This completes the proof.
• Where can I download a graphing calculator for my computer like what Sal is using?
• desmos.com has a free, online graphing/scientific calculator that I use often. Even though you can make an account on the site, you don't need to to use their calculator.
• So why do we always take log(base 10) ?
Why we dont take any other number ?
• You can do that. In fact base 10, base 2 and base e (natural logarithm) are only the most common.
The limitation is your calculator. Simple calculators only have a function for base 10 and base e and that's it.
• I didn't get that last part of the video, where Clog(a) = log(b) turns into the very last part of what Khan was writing about, logb/loga. How does that work out? Please and thank you :)
• Only correcting srerm5:
C x log(a) = log (b)
C = log(b) / log(a)
• How can you do this WITHOUT a calculator?
• I believe you could use a slide rule (if you have one), or consult one of many charts of logs drawn up by mathematicians of old
• How would i solve 3^(x − 4) = 6 using the change of base formula.
• 3^(x-4) = 6

(3^x)/(3^4) = 6 by the exponent properties

3^x = 6*81 multiply both sides by 3^4 or 81

log_3(486)=x by converting exponent for to log form. Thsi reads as log base 3 of 486

Fromt here you can make it log(486)/log(3) where the logs are any base you want.
• Can you put any number as a base in the logarithmic base conversion formula?
• No. The base of the base conversion formula is always 10. The reason for the equation is to allow you to enter a log with a non 10 base into a calculator.
The formula is:
log base d of c = (log base 10 of c)/(log base 10 of d)
• Why does clogx(a)=logx(a^c)?
• One way to think about it is this. Exponentiation is repeated multiplication. That is a^c = a · a · ... · a, where a is repeated c times.
Therefore log(a^c) = log(a · a · ... · a)
Now log(a · b) = log(a) + log(b),
so log(a · a · ... · a) = log(a) + log(a) + ... + log(a) (repeated c times)
= c·log(a)