If `log_(b)5=a,log_(b)2.5=c,and5^(x)=2.5`, then x=

Options

A. ac
B. `(c)/(a)`
C. `a+c`
D. `c-a`

Explanation

To solve the equation given in the problem, we will follow these steps:

### Step 1: Rewrite the logarithmic equations
We start with the logarithmic equations provided:
1. \( \log_b 5 = a \) implies \( b^a = 5 \) (Equation 1)
2. \( \log_b 2.5 = c \) implies \( b^c = 2.5 \) (Equation 2)

### Step 2: Use the given equation
We are also given that:
\[ 5^x = 2.5 \]

### Step 3: Substitute the values from Equations 1 and 2
We can substitute the values of \( 5 \) and \( 2.5 \) from Equations 1 and 2 into the equation \( 5^x = 2.5 \):
\[ (b^a)^x = b^c \]

### Step 4: Apply the power of a power property
Using the property of exponents that states \( (x^m)^n = x^{m \cdot n} \), we can rewrite the left side:
\[ b^{a \cdot x} = b^c \]

### Step 5: Set the exponents equal to each other
Since the bases are the same (both are \( b \)), we can set the exponents equal to each other:
\[ a \cdot x = c \]

### Step 6: Solve for \( x \)
To isolate \( x \), we divide both sides by \( a \):
\[ x = \frac{c}{a} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{c}{a} \]

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