Question:easy

If \[ \log_a 2+\log_a 5=1, \] then the value of \(a\) is:

Show Hint

If \[ \log_a b=1, \] then immediately conclude that \[ a=b, \] because \[ a^1=b. \] This shortcut frequently appears in logarithmic equations.
Updated On: Jun 10, 2026
  • \(7\)
  • \(10\)
  • \(12\)
  • \(15\)
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Write the given equation.
We have $\log_a 2+\log_a 5=1$. Both logarithms share the same base $a$.

Step 2: Combine the two logarithms.
A log rule says $\log_a m+\log_a n=\log_a(mn)$. So adding logs of $2$ and $5$ gives the log of their product.

Step 3: Multiply the numbers.
Since $2\times5=10$, the equation becomes \[ \log_a 10=1. \]

Step 4: Switch to exponential form.
The statement $\log_a 10=1$ means $a$ raised to the power $1$ equals $10$. That is, \[ a^1=10. \]

Step 5: Read off $a$.
Since $a^1=a$, we directly get $a=10$.

Step 6: Quick check.
With $a=10$, $\log_{10}2+\log_{10}5=\log_{10}10=1$, which matches. So the value is correct. \[ \boxed{10} \]
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