Solve the following equations.`x^((log_10x+5)/3)=10^(5+log_10x)`

Answer

To solve the equation \( x^{\frac{\log_{10} x + 5}{3}} = 10^{5 + \log_{10} x} \), we will follow these steps:

### Step 1: Rewrite the equation in logarithmic form
We start with the given equation:
\[ x^{\frac{\log_{10} x + 5}{3}} = 10^{5 + \log_{10} x} \]
Taking the logarithm (base 10) of both sides:
\[ \log_{10}\left(x^{\frac{\log_{10} x + 5}{3}}\right) = \log_{10}\left(10^{5 + \log_{10} x}\right) \]

### Step 2: Apply the power rule of logarithms
Using the property \( \log(a^b) = b \log(a) \):
\[ \frac{\log_{10} x + 5}{3} \cdot \log_{10} x = 5 + \log_{10} x \]

### Step 3: Clear the fraction
Multiply both sides by 3 to eliminate the fraction:
\[ (\log_{10} x + 5) \log_{10} x = 3(5 + \log_{10} x) \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ \log_{10} x \cdot \log_{10} x + 5 \log_{10} x = 15 + 3 \log_{10} x \]
This simplifies to:
\[ (\log_{10} x)^2 + 5 \log_{10} x - 3 \log_{10} x - 15 = 0 \]
Which can be rearranged to:
\[ (\log_{10} x)^2 + 2 \log_{10} x - 15 = 0 \]

### Step 5: Factor the quadratic equation
Now we will factor the quadratic:
\[ (\log_{10} x + 5)(\log_{10} x - 3) = 0 \]

### Step 6: Solve for \(\log_{10} x\)
Setting each factor to zero gives us:
1. \( \log_{10} x + 5 = 0 \) → \( \log_{10} x = -5 \) → \( x = 10^{-5} \)
2. \( \log_{10} x - 3 = 0 \) → \( \log_{10} x = 3 \) → \( x = 10^{3} \)

### Step 7: Write the final solutions
Thus, the solutions to the equation are:
\[ x = 10^{-5} \quad \text{or} \quad x = 10^{3} \]