if (logx to the base k)(logk to the base 5)=log5 to the base x,k is not equal to 1,k>0,then x is equal to? plzz do show working steps also

We are given the equation involving logarithms:

Given:

(log_kx) × (log₅k) = log_x5

We need to find the value of x, given that k ≠ 1 and k > 0.

Step-by-Step Solution

Step 1: Apply Change of Base Formula

We convert all logarithms to a common base (say base 10 or base e), using the formula:

log_ab = log b / log a

So each term becomes:

• log_kx = log x / log k
• log₅k = log k / log 5
• log_x5 = log 5 / log x

Step 2: Substitute into the Equation

(log x / log k) × (log k / log 5) = log 5 / log x

Step 3: Simplify Left-Hand Side

(log x × log k) / (log k × log 5) = log 5 / log x

The log k terms cancel out:

log x / log 5 = log 5 / log x

Step 4: Cross Multiply

(log x)² = (log 5)²

Step 5: Take Square Root on Both Sides

log x = ± log 5

This gives us two possible cases:

• log x = log 5 → x = 5
• log x = –log 5 → x = 1/5

Step 6: Check Validity

Recall that the original equation involves log_kx, so both x = 5 and x = 1/5 are valid as long as they are positive, which they are. We now check both:

Case 1: x = 5

LHS = log_k5 × log₅k = (log 5 / log k) × (log k / log 5) = 1

RHS = log₅5 = 1 ✓

Case 2: x = 1/5

LHS = log_k(1/5) × log₅k = (log(1/5) / log k) × (log k / log 5) = log(1/5) / log 5

RHS = log_1/55 = log 5 / log(1/5)

So LHS ≠ RHS in this case because one is the reciprocal of the other. ❌

Final Answer:

x = 5