Prove that log base a of x divided by log base a/b of x is equal to 1+log base a of(1/b)
Where do I start?
since
log_u(x)/log_v(x) = log_v(u) and
log_u(x) = 1/log_x(u), we have
log_a/b(a) = 1/log_a(a/b)
= log_a(a) + log_a(1/b)
= 1 + log_a(1/b)
If those two properties of logs have not yet been delivered to you, work to convince yourself that they are true. I'm sure you can find many proofs online.
To prove the given equation, we can start by using the properties of logarithms.
Let's break down the equation step by step:
1. Start with the left-hand side of the equation: log base a of x divided by log base a/b of x.
2. Recall the logarithmic property that states log base a of x divided by log base a of y is equal to log base y of x. Applying this property, we can rewrite the left-hand side as log base a/b of x.
3. Convert log base a/b of x to a single logarithm using the logarithmic change of base formula. This formula states that log base c of x can be rewritten as log base a of x divided by log base a of c. In our case, we can rewrite log base a/b of x as log base a of x divided by log base a of (a/b).
4. Simplify the expression log base a of (a/b). Apply the same logarithmic property as in step 2 to log base a of (a/b), which gives us log base b of a.
5. Substitute log base a of (a/b) with log base b of a in the expression from step 3.
6. We get log base a of x divided by log base a of log base b of a.
7. Now, apply another logarithmic property, which states that log base a of log base b of x is equal to 1 divided by log base b of a. Using this property, we can rewrite the expression from step 6 as log base a of x multiplied by log base b of a.
8. Simplify this expression to log base b of a times log base a of x.
9. Finally, rearrange the order of the logarithms using the commutative property of multiplication, giving us log base a of x multiplied by log base b of a.
10. Note that log base a of x multiplied by log base b of a is equal to log base a of x plus log base a of b.
11. Substitute this simplified expression back into the original equation from step 1, and we get log base a of x plus log base a of b.
12. Now, we can rewrite log base a of b as log base a of (1/a), since b is equal to 1 divided by a.
13. Simplify log base a of (1/a) to -1 using the logarithmic property that states log base a of 1/a is equal to -1.
14. Substitute this simplified expression back into the equation, and we have log base a of x plus (-1), which simplifies to log base a of x minus 1.
15. Finally, we obtain the right-hand side of the equation, which is equal to 1 plus log base a of (1/b).
Therefore, the left-hand side of the equation is equal to the right-hand side, and we have proven that log base a of x divided by log base a/b of x is equal to 1 plus log base a of (1/b).