Question

Solve this problem:
㏒₅₄168=?
If:
a = ㏒₇12
b = ㏒₁₂24

Answer 1

㏒₅₄168 = (1+ab)/(8a-5ab)

Explanation:

You want ㏒₅₄168 written in terms of a and b, where ...

• a = ㏒₇12
• b = ㏒₁₂24

Log expressions

The rules of logarithms are relevant here:

`\log_b(a)=(\log(a))/(\log(b))\\\\\log(ab)=\log(a)+\log(b)`

In order to simplify the writing in what follows, we want to define ...

• p = log(2)
• q = log(3)
• r = log(7)

Now, we can write the equations ...

a = log(12)/log(7) = (2p+q)/r . . . . . . . . . [eq1]

b = log(24)/log(12) = (3p+q)/(2p+q) . . . [eq2]

Using these, we can solve for p and q:

2p +q = ar . . . . . . . . . . . . . [eq3], from [eq1]

(3 -2b)p +(1 -b)q = 0 . . . . . [eq4] from [eq2]

Using your favorite solution method for linear equations, you find ...

p = ar(1 -b)/(2(1 -b) -(3 -2b)) = ar(1 -b)/(2 -2b -3 +2b) = ar(b -1)

q = (-ar(3 -2b)/(-1) = ar(3 -2b)

Desired logarithm

Substituting these expressions into ...

`log_54(168)=\frac{\log{(2^3\cdot3\cdot7)}}{\log{(2\cdot3^3)}}=(3p+q+r)/(p+3q)\\\\=(3ar(b-1)+ar(3-2b)+r)/(ar(b-1)+3ar(3-2b))=(3ab-3a+3a-2ab+1)/(ab-a+9a-6ab)\\\\=\boxed{log_54(168)=(1+ab)/(8a-5ab)}`

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Additional comment

The calculator confirms the result.