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In mathematics, the change of base formula is a handy tool for handling logarithms with bases that are not so conveniently dealt with. The formula is expressed as \( x = \log_b a = \frac{\log_c a}{\log_c b} \), where:

• \( a \) is the number you are taking the log of,
• \( b \) is the base of the logarithm you wish to convert from,
• \( c \) is the base to which you are converting, often \( c = 10 \) for common logarithms.

Understanding this formula allows us to simplify and calculate logarithms in any base by converting them into a more manageable form.
To explain using our example, we have \( 4^x = 3 \). We wish to solve for \( x \) using base 10 logarithms. By applying the formula, \( x = \frac{\log_{10} 3}{\log_{10} 4} \), enabling us to find \( x \) when the base is not 10 or \( e \). Step by step, each of these logarithms can be calculated using a calculator, revealing the approximate value for \( x \).
The change of base formula is not only pivotal in theoretical problems, but also everyday calculations where precision is key, especially when dealing with exponential growth and decay.

Finding the x-intercept of a function is crucial because it pinpoints where the graph of the function crosses the x-axis, providing insights into the behavior of the graph. To locate the x-intercept, one needs to solve the equation \( f(x) = 0 \). This involves setting the entire function to zero and solving for \( x \).
For the given function \( f(x) = 4^x - 3 \), setting \( 4^x - 3 = 0 \) and solving yields \( 4^x = 3 \). From here, using the change of base formula, we then solve for \( x \), giving us \( x \approx 0.792 \) with the help of a calculator.
Understanding where the x-intercept lies helps in sketching the graph and visualizing how the function behaves as it traverses the x-axis. The x-intercept can often represent real-world scenarios like break-even points in economics or the point where a population reaches equilibrium in biology.

Sketching a graph provides a visual representation of a function, making it easier to interpret and analyze. For an exponential function like \( f(x) = 4^x - 3 \), there are some key features to keep in mind when sketching:

• **Exponential Growth:** The graph rises steadily from left to right, showcasing growth as \( x \) increases.
• **Vertical Shift:** Due to the \(-3\), the entire graph is shifted 3 units downward.
• **Asymptote:** There's a horizontal line at \( y = -3 \) that the graph approaches but never touches.
• **Key Points:** Identify and plot points like where the graph passes through \( (0, 1) \), calculated by \( f(0) = 4^0 - 3 = 1 \).
• **X-intercept:** Marking the x-intercept at \( x \approx 0.792 \) helps confirm the crossing point on the x-axis.

With these in mind, your sketch will serve as a valuable tool in understanding how the function acts across different values of \( x \). Breaking it down in simple steps ensures that nothing is overlooked, and you fully comprehend the function's behavior visually.