Question

compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x. for what values of x is f=g, and f>g, and f

Answer 1

The graphs for both equations are shown in the figure. The blue curve represents the f(x) function while the orange curve represents the g(x) function. Based on the figure the function of f equals g when x is less than zero. The function of f will be greater than g when x is greater than zero.

Thus,

f=g, if x<0 and f>g, if x>0

Answer 2

This problem has several items. So, let's solve it step by step.

1. Compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x.

In the Figure below, we have tha graph of the two functions. The graph in red is $f(x)=log(7x)$ and the graph in blue is $g(x)=log(4x)$. The x-intercept of $f$ is:

$y=log(7x) \\ \\ 0=log(7x) \\ \\ 10^{0}=10^{log(7x)} \\ \\ 1=7x \\ \\ x=\frac{1}{7}=0.14$

On the other hand, the x-intercept of $g$ is:

$y=log(4x) \\ \\ 0=log(4x) \\ \\ 10^{0}=10^{log(4x)} \\ \\ 1=4x \\ \\ x=\frac{1}{4}=0.25$

Each graph begins in the fourth quadrant and is increasing quickly. As the graph crosses the x-axis at each x-intercept, each graph does not increase as fast. The graph continues to increase slowly throughout the first quadrant.

2. For what values of x is f=g

We can find this answer by taking this equation:

$f(x)=g(x) \\ \\ log(7x)=log(4x) \\ \\ 10^{log(7x)}=10^{log(4x)} \\ \\ 7x=4x \\ \\ 7=4$

As you can see this is an absurd result since 7 is not equal to 4. The conclusion is that the function $f$ is always different from $g$, that is, $f\neq g \ always!$

3. For what values f>g

From the graph, we can see that the red function is always greater than the blue function. Therefore, $f>g \ Always!$