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The problem asks to find the equation of the tangent line \( l \) to the curve \( f(x) = 10^x \) at the point where \( x = \log 7 \).

Let's go through the steps to solve this:

### 1. Find the slope of the tangent line

To find the slope of the tangent line, we need to compute the derivative of the function \( f(x) = 10^x \).

Using the derivative rule for exponential functions:
\[
\frac{d}{dx}(10^x) = 10^x \ln(10)
\]
So, the derivative \( f'(x) = 10^x \ln(10) \).

The slope of the tangent line at \( x = \log 7 \) is:
\[
f'(\log 7) = 10^{\log 7} \ln(10)
\]
Since \( 10^{\log 7} = 7 \), the slope becomes:
\[
f'(\log 7) = 7 \ln(10)
\]

### 2. Find the point of tangency

The point of tangency is on the curve \( f(x) = 10^x \) when \( x = \log 7 \). We know:
\[
f(\log 7) = 10^{\log 7} = 7
\]
So the point of tangency is \( (\log 7, 7) \).

### 3. Equation of the tangent line

Using the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) = (\log 7, 7) \), we substitute:
\[
y - 7 = 7 \ln(10)(x - \log 7)
\]
Expanding the equation gives:
\[
y = 7 \ln(10)(x - \log 7) + 7
\]

This is the equation of the tangent line in the form \( y = mx + b \).

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Do you need further clarifications or details?

Here are 5 related questions you could explore:
1. How do you find the slope of the tangent line for other exponential functions?
2. What is the derivative of other logarithmic base exponential functions, such as \( e^x \)?
3. How does the point-slope form compare to the slope-intercept form?
4. How would you apply this to finding tangent lines for other values of \( x \)?
5. What are the implications of changing the base of the logarithm for the equation of the tangent line?

**Tip**: To find tangent lines, always compute the derivative first to get the slope, then use the point-slope formula for the line!

Find the equation of the tangent line to the curve f(x) = 10^x at x = log 7, using the form y = mx + b.

Find the equation of the tangent line to the curve f(x) = 10^x at x = log 7. This solution covers concepts from calculus, such as derivatives of exponential functions and the point-slope form for tangent lines. Suitable for advanced high school students.

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