The solution of the equation `5^(log_(a)x)+5x^(log_(a)5)=3, (agt0)` is

Options

A. `a^(-log_(5)2)`
B. `a^(log_(5)2)`
C. `2^(-log_(5)a)`
D. `2^(log_(5)a)`

Answer

To solve the equation \( 5^{\log_a x} + 5x^{\log_a 5} = 3 \) where \( a > 0 \), we can follow these steps:

### Step 1: Rewrite the equation using properties of logarithms
We know that \( \log_a x = \frac{\log x}{\log a} \) and \( \log_a 5 = \frac{\log 5}{\log a} \). Thus, we can rewrite the equation as:
\[ 5^{\frac{\log x}{\log a}} + 5x^{\frac{\log 5}{\log a}} = 3 \]

### Step 2: Simplify the terms
Using the property \( b^{\log_b x} = x \), we can rewrite \( 5^{\log_a x} \) as \( x^{\log_a 5} \):
\[ x^{\log_a 5} + 5x^{\log_a 5} = 3 \]

### Step 3: Factor the equation
Let \( k = \log_a 5 \). Then the equation simplifies to:
\[ x^k + 5x^k = 3 \]
This can be factored as:
\[ (1 + 5)x^k = 3 \]
or
\[ 6x^k = 3 \]

### Step 4: Solve for \( x^k \)
Dividing both sides by 6 gives:
\[ x^k = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Solve for \( x \)
Now, we can express \( x \) in terms of \( k \):
\[ x = \left(\frac{1}{2}\right)^{\frac{1}{k}} = \left(\frac{1}{2}\right)^{\frac{\log a}{\log 5}} \]

### Step 6: Final expression for \( x \)
Using the properties of exponents:
\[ x = a^{\frac{\log(1/2)}{\log 5}} = a^{-\frac{\log 2}{\log 5}} \]

Thus, the solution to the equation \( 5^{\log_a x} + 5x^{\log_a 5} = 3 \) is:
\[ x = a^{-\frac{\log 2}{\log 5}} \]