compare the graphs of the logarithmic functions f(x)=log7x and g(x)=log4x for what values of x is f=g f>g and f<g explain how you know
log(1) = 0 for any base
I assume in your functions that 4 and 7 are the bases.
Since 4^x < 7^x, log4x > log7x for x>1
log4x < log7x for x<1
recall that if y = log_b(x) then
dy/dx = 1/lnb * 1/x
To compare the graphs of the logarithmic functions f(x) = log7x and g(x) = log4x, we need to analyze their behavior in terms of equality, f > g, and f < g.
1. Equality (f = g):
To find the values of x where f(x) = g(x), we set the two logarithmic expressions equal to each other:
log7x = log4x.
To solve this equation, we can use the property of logarithms that states if loga(b) = loga(c), then b = c. Thus, we have:
7x = 4x.
Dividing both sides by 4x (assuming x > 0), we get:
7/4 = 1.
Since 7/4 ≠ 1, there is no value of x where f(x) = g(x).
2. f > g:
To determine when f(x) > g(x), we need to compare the bases of the logarithms.
Given f(x) = log7x and g(x) = log4x, we can see that 7 > 4. Therefore, for any given value of x, f(x) will always be greater than g(x).
In other words, f(x) > g(x) for all x > 0.
3. f < g:
As stated above, f(x) will always be greater than g(x) since 7 > 4. Therefore, there are no values of x for which f(x) < g(x).
To summarize:
- There are no values of x where f(x) = g(x).
- f(x) > g(x) for all x > 0.
- There is no value of x where f(x) < g(x).
To compare the graphs of the logarithmic functions f(x) = log7x and g(x) = log4x, we need to analyze their properties and determine for what values of x they are equal (f=g), when f is greater than g (f>g), and when f is less than g (f<g).
First, let's look at the base of the logarithms. In f(x) = log7x, the base is 7, while in g(x) = log4x, the base is 4. Since 7 > 4, we can conclude that logarithms with a base of 7 grow faster than those with a base of 4.
Now, let's analyze the different cases:
1. f(x) = g(x): This means f and g are equal for certain values of x. To find these values, we can set the two logarithmic functions equal to each other:
log7x = log4x
To solve this equation, we can apply the property of logarithms that states if two logarithmic expressions with the same base are equal, then their arguments are equal:
7x = 4x
Dividing both sides of the equation by 4x, we get:
7/4 = 1
Since 7/4 ≠ 1, we conclude that there are no values of x for which f(x) = g(x). Therefore, f is never equal to g.
2. f(x) > g(x): This means f is greater than g for certain values of x. Since logarithms with a greater base grow faster, we can conclude that f(x) > g(x) for all positive values of x. In other words, f(x) > g(x) for x > 0.
3. f(x) < g(x): This means f is less than g for certain values of x. Similar to the previous case, logarithms with a greater base grow faster. Therefore, f(x) < g(x) when x < 0.
To summarize:
- f(x) = g(x) for no values of x.
- f(x) > g(x) when x > 0.
- f(x) < g(x) when x < 0.
By understanding the properties of logarithms and their behavior with respect to the base, we can determine how the graphs of logarithmic functions compare for different values of x.