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

Lesson 26: Logarithmic equations (Algebra 2 level)

# Logarithmic equations: variable in the argument

Sal solves the equation log(x)+log(3)=2log(4)-log(2). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Where are logs used in real life?
• The human ear works as a logarithmic function. The tempered musical scale is exponential so after passing through a logarithmic function (ear) it become linear. This mix of functions makes the transition from notes of the scale perceived by our brain softly as if the notes were located exactly one after the other. Basically, the frequencies of the musical notes are equally logarithmic scaled.
http://en.wikipedia.org/wiki/Music_and_mathematics
• So does order of operation still have to be followed with logarithms? Here's why I ask this:

2*log(4) - log(2)

What I did was I first used the division property and I got 2*log(4/2) = 2*log(2). Only after this I moved the 2 in front to be the exponent of log(2) so I got log(4).

In the video Sal first multiplied and then divided the logarithm, resulting in log(8).

Have I done something wrong? Thanks.
• Order of operations still apply, but remember the properties, in this case

a*log_b(c)=log_b(c)^a

2*log(4)-log(2)
log(4)^2-log(2)
log(16)-log(2)
log(16/2)
log(8)

Go learn yourself the properties, they are very useful :)
• I'm getting confused at , when he offers an alternate way of solving the equation. Can someone theoretically explain to me how raising both sides of the equation to the log's base solves the problem?
I understand the lesson up until that part.
• logarithms are just inverse functions of exponential functions so that the base and the exponents cancel and equal 1 .try this logany base (withthat number)=1
as well exponets leading coeffitient with raised with any logsame numbe =1

let say 10^x(power)=100 by logarithm rules it inverse it intern of x
log(10_base)(100)=x so that x=2
log( 10^x(power))=log(100) this simplifies to x=log 100 or 2
• How do you solve a logarithmic equation with the same base on both sides?
Example: loga(b+1) = loga(c+1)
Thanks!
• ... Don't you think `b` is equal to `c` at the first sight?
• When solving for a logarithm, would you need to find the domain of the equation as well? (To ensure that the solution is within the domain of the equation)
• Yes you would. Same for radical equations.
• What is y=b to the 2nd power
• What if instead of having log_ throughout the equation, you have two logarithms on one side but just a number on the other side of the equal sign?
Ex]
log base of 9 (x -7) plus log base of 9 (x - 7) is equal to one.
Would you multiply the (x - 7) with each other and then equal it to one?
• Here's how to solve that kind of problem. Let me know if you don't understand any step and I'll try to explain in greater detail.
log₉(x-7) + log₉(x-7) = 1
2log₉(x-7) = 1
log₉(x-7) = ½
9^[log₉(x-7)] = 9^(½)
x-7 = 3
x = 10
Here is a similar but slightly more difficult problem:
log₉(x-4) + log₉(x+4) = 1
log₉{(x-4)(x+4)} = 1
9^[log₉{(x-4)(x+4)} ] = 9^1
(x-4)(x+4) = 9
x²-16 = 9
x² = 25
x = 5
Note: we have to discard any values for x that would lead to a log with an argument that is 0 or a negative number, that is why I only used the principle square roots. In some occasions, you may need to use the negative square roots -- it all depends on what keeps the argument a positive number.
• so if i had, 3log(base2)x - log(2)(x-2) = 4
ive really got log2(x^3/x-2) = 4?
• That looks right, yes. Further, you have 2^4, or 16, equal to x^3/(x-2).
• I'm slightly confused at of the video. When i got to the stage of: log3x = log 8
i divided log3 on both sides leaving:

x = log8 / log3

I didn't drop the log on both sides as he did in the video, and when i put it in the calculator, they're two different answers. Is dividing out by log3 agebraicly incorrect or is it just assumed you have to drop the log when they're on both sides of the equation in the above situation?

Thanks so much for the video though! It was super helpful :)
• Yes, when you have the same log base on both sides of the equation (log base 10 or log base 2) they will cancel each other out. It's like if you had x+2=x+2. The x's will cancel out when you subtract the x's from each other. When solving logs, many of the same rules for solving equations apply, the only difference is in addition to solving for x, you have to condense the logs.
(1 vote)
• When you have constants on each side of the equation and they are being raised to binomial powers, how do you solve?
For example: 5^(x+3) = 7^(2x+3)
``5^(x+3) = 7^(2x+3) ln [ 5^(x+3)] = ln [7^(2x+3) ](x+3) ln (5) = (2x+3) ln (7)Now use the distributive property:x ln( 5) + 3 ln 5 = 2x ln(7) + 3 ln(7)Rearrange to get x on the left hand side:x ln 5 - 2x ln 7 = 3 ln 7 - 3 ln 5Factoring gives us:x (ln 5 - 2 ln 7) = 3 (ln 7 - ln 5)x = 3 (ln 7 - ln 5) / (ln 5 - 2 ln 7)``