Logarithms
By DarthVader
Date: 2021-12-19
Topic: 90 see comments
Post views: 1249
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Logarithms
Definition:
loga x = y ⇢ ⇢ ay = x
Rules:
logx (ab) = logx a + logx b
logx (a/b) = logx a − logx b
logx (uc) = c × logx u
logx (1/b) = -logx b
logx 1 = 0
logx x = 1
hint - you can't take the log of 0
Video:
https://www.youtube.com/watch?v=fnhFneOz6n8&t=150s
Strategy:
First look at the base number and work out which number to raise it to to get the logarithm
Example Problems
Problem 1:
log3 (x − 4) = 2
Step 1:
The opposite of a logarithm is an exponential.
Raising the entire left side of this equation to the same base number as the logarithm, will cancel out the logarithm.
Don't forget to do the same on both sides:
3log3 (x − 4) = 32
Step 2:
The exponent and the logarithm will cancel eachother out on the left side of the equation:
3log3 (x − 4) = 32
Step 3:
Now we have:
(x − 4) = 32
Step 4:
3 to the 2nd power equals 9:
(x − 4) = 9
Step 5:
Get x by itself by removing the -4 by adding 4 to both sides:
x = 13
Problem 2:
log5 x2 = −2
Step 1:
The opposite of a logarithm is an exponential.
Raising the entire left side of this equation to the same base number as the logarithm, will cancel out the logarithm.
Don't forget to do the same on both sides:
5log5 x2 = 5−2
Step 2:
The exponent and the logarithm will cancel eachother out on the left side of the equation:
5log5 x2 = 5−2
Step 3:
Now we have:
x2 = 5−2
Step 4:
5−2 just equals 1/52 which just equals 1/25 so now we have:
x2 = 1/25
Step 5:
Now to get rid of the exponent we take the square root and do the same to both sides:
√x2 = √1/25
Step 6:
The square root and the exponent will cancel eachother out on the left side:
√x2 = √1/25
Step 7:
Now we have:
x = √1/25
Step 8:
The square root of 1 is 1, and the square root of 25 is 5, so:
x = 1/5
More examples:
Problem 1:
Step 1:
loga a = _? ⇥ log of a with base a equals what?
Step 2:
a? = a ⇥ base a, raised to what, equals a?
Step 3:
a1 = a ⇥ base a raised to the power of one equals a
Step 4:
loga a = 1 ⇥ so the answer is 1
Problem 2:
Step 1:
loga 1 = _? ⇥ log of 1 with base a equals what?
Step 2:
a? = 1 ⇥ base a, raised to what, equals 1?
Step 3:
a0 = 1 ⇥ any number raised to 0 equals 1
Step 4:
loga 1 = 0 ⇥ so the answer is 0
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