https://file.aiquickdraw.com/mathbot/file/1729602029326k7qo6mpk.jpg

The equation in the image is:

\[
3 \log_4{x} + 3 \log_5{x} = 1
\]

We need to solve for \( x \).

### Step 1: Simplify the equation
We can factor out the 3 on both sides of the equation:
\[
3(\log_4{x} + \log_5{x}) = 1
\]

Divide both sides by 3:
\[
\log_4{x} + \log_5{x} = \frac{1}{3}
\]

### Step 2: Express the logarithms in terms of natural logarithms
We can rewrite the logarithms using the change of base formula:
\[
\log_4{x} = \frac{\ln{x}}{\ln{4}}, \quad \log_5{x} = \frac{\ln{x}}{\ln{5}}
\]

Substitute these into the equation:
\[
\frac{\ln{x}}{\ln{4}} + \frac{\ln{x}}{\ln{5}} = \frac{1}{3}
\]

### Step 3: Combine the terms
Factor out \( \ln{x} \) on the left-hand side:
\[
\ln{x} \left( \frac{1}{\ln{4}} + \frac{1}{\ln{5}} \right) = \frac{1}{3}
\]

Now calculate the sum of the constants:
\[
\frac{1}{\ln{4}} + \frac{1}{\ln{5}} = \frac{1}{\ln{2^2}} + \frac{1}{\ln{5}} = \frac{1}{2 \ln{2}} + \frac{1}{\ln{5}}
\]

Let’s approximate the values of \( \ln{2} \) and \( \ln{5} \):
\[
\ln{2} \approx 0.6931, \quad \ln{5} \approx 1.6094
\]

Substitute these into the equation:
\[
\frac{1}{2 \times 0.6931} + \frac{1}{1.6094} \approx 0.7213 + 0.6213 = 1.3426
\]

### Step 4: Solve for \( \ln{x} \)
Now, substitute back into the equation:
\[
\ln{x} \times 1.3426 = \frac{1}{3}
\]

Solve for \( \ln{x} \):
\[
\ln{x} = \frac{1}{3 \times 1.3426} \approx \frac{1}{4.0278} \approx 0.2483
\]

### Step 5: Solve for \( x \)
Now, exponentiate both sides to solve for \( x \):
\[
x = e^{0.2483} \approx 1.282
\]

Thus, \( x \approx 1.282 \).

Would you like more details on any of the steps? Here are some related questions you could explore:

1. How can I solve logarithmic equations with different bases?
2. What is the change of base formula for logarithms, and how is it used?
3. How can logarithmic properties simplify complex equations?
4. How do you convert natural logarithms to base 10 or another base?
5. What are the common uses of logarithmic functions in real-life applications?

**Tip**: When solving logarithmic equations, try expressing all logs in the same base to simplify calculations.

Für welches x gilt 3 log_4(x) + 3 log_5(x) = 1?

This step-by-step solution explains how to solve the logarithmic equation 3log_4(x) + 3log_5(x) = 1 using the change of base formula and properties of logarithms. Learn the mathematical steps to simplify and solve the equation, suitable for high school students.

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