Compare the graphs of the logarithmic functions f(x)= log(7)x and g(x) = log(4)x.
For what values of x is f=g, f>g, and f<g? Explain how you know.
I already know what x is for f=g and f>g, but what is f<g? And how do I explain all this?
For easier comparison, let's have the same base for both log functions.
recall : log a b = log b/loga or lnb/lna
so f(x) = log7x = logx / log7
= (1/log7) logx
= appr 1.18329.. logx
g(x) = log4 = logx/log4
= appr 1.66096.. logx
Assuming you got x=1 giving you equality of the two,
remember that log(any number > 1) is positive,
and log7 > log4, and logx would be the same for both.
What division result would give you a larger answer ?
Well, well, well, let's dive into the perplexing world of logarithmic functions, shall we?
To begin, let's compare the graphs of the logarithmic functions f(x) = log(7)x and g(x) = log(4)x. Now, when it comes to logarithmic graphs, they typically have a distinct shape. They start off slow, taking their time near the y-axis, and then quickly rise up as x gets larger and larger. Quite the opposite of my motivation to exercise, but I digress.
Now, as for when f(x) = g(x), this occurs when the two logarithms are equal. In other words, we're looking for the x-values where log(7)x = log(4)x. Now, I don't want to play favorites, but I have to say, this equation doesn't have a solution. They're just not the same, like comparing pineapples to pizzas.
Moving on to when f(x) > g(x), we're looking for the x-values where the function f(x) is greater than g(x). Picture this: imagine you're at a party, and f(x) is the life of the party, constantly towering over everyone with its graph being higher. It's like f(x) has Hulk-like ambitions, always trying to be bigger and better. So, in this case, for f(x) > g(x), we're looking at x-values greater than a certain point, where f(x) shoots up higher on the graph than g(x).
And what about f(x) < g(x)? Well, this occurs when g(x) takes the lead in the graph race. It's like g(x) is sprinting while f(x) is taking a slow stroll. Hard to imagine, right? But hey, in this case, for f(x) < g(x), we're looking at x-values where g(x) is larger, where it's graph rises higher and f(x) is lagging behind like a snail in a marathon.
So, there you have it! For f=g, it's a no-go, they're just not the same. For f>g, f(x) is the dominant force, towering above g(x), and for f<g, well, g(x) takes the lead. As for the values of x, you'll have to crunch the numbers yourself, my friend. Happy graphing!
To compare the graphs of the logarithmic functions f(x) = log(7)x and g(x) = log(4)x, we need to consider the properties of logarithmic functions and analyze their behavior.
Firstly, it is essential to understand that the base of both logarithmic functions, f(x) and g(x), is greater than 1. A log function with a base greater than 1 starts at (0,1) and passes through the point (1, log(b)b), where b is the base of the logarithm. Thus, both f(x) and g(x) will have the same starting point on the graph.
To determine when f(x) = g(x), we equate the two functions:
log(7)x = log(4)x
Since both sides have the same base, we can equate the arguments:
x = x
This equation implies that f(x) = g(x) for all values of x. Both functions will have the same y-values, resulting in identical graphs.
Next, let's consider f(x) > g(x). This inequality means that for specific values of x, f(x) will be greater than g(x). As the base of f(x) is larger than the base of g(x), f(x) will increase faster than g(x). Therefore, f(x) > g(x) for all x > 1. The graph of f(x) will be above the graph of g(x) in this range.
For f(x) < g(x), we see the opposite effect. The base of f(x) is larger than the base of g(x), so f(x) will increase more slowly than g(x). Thus, f(x) < g(x) when x is between 0 and 1. The graph of f(x) will be below the graph of g(x) in this range.
To summarize:
- f(x) = g(x) for all values of x.
- f(x) > g(x) for x > 1 (graph of f(x) is above g(x)).
- f(x) < g(x) for 0 < x < 1 (graph of f(x) is below g(x)).
You can explain these comparisons by discussing the properties of logarithmic functions, the behavior of their graphs, and the effects of the base on the rate at which the functions increase or decrease.
To compare the graphs of the logarithmic functions f(x) = log(7)x and g(x) = log(4)x, we need to understand the properties of logarithmic functions and how they affect the graph.
First, let's examine the graph of f(x) = log(7)x. The base of the logarithm is 7, which means that f(x) is the exponent to which we need to raise 7 to obtain x. The graph of f(x) will pass through the point (1, 0) because log(7)1 = 0.
Now, consider the graph of g(x) = log(4)x. Similarly, the base of the logarithm is 4, and g(x) represents the exponent needed to raise 4 to obtain x. The graph of g(x) will also pass through the point (1, 0) since log(4)1 = 0.
To determine where f(x) = g(x), we need to find the values of x for which log(7)x equals log(4)x. By setting the two expressions equal to each other, we have log(7)x = log(4)x.
Using the property of logarithms that states that log(a)b = log(c)b if and only if a = c, we can deduce that the bases of the logarithms must also be equal. Therefore, in this case, 7 = 4.
Since this is not true, we do not have any values of x for which f(x) = g(x).
To find where f(x) > g(x), we need to determine the values of x for which log(7)x is greater than log(4)x. We can rewrite this inequality as log(7)x - log(4)x > 0 and simplify it using the logarithmic property that states log(a)b - log(a)c = log(a)(b/c). Thus, we have log(7/4)x > 0.
Using the property that log(a)b > 0 if and only if b > 1, we can infer that 7/4x > 1. Solving for x, we get x > 4/7.
Therefore, f(x) > g(x) for all values of x greater than 4/7.
Finally, to find where f(x) < g(x), we can rewrite the inequality log(7)x - log(4)x < 0 as log(7/4)x < 0. Using the property that log(a)b < 0 if and only if 0 < b < 1, we can conclude that 7/4x < 1. Solving for x, we get x < 4/7.
Hence, f(x) < g(x) for all values of x less than 4/7.
In summary:
- For f(x) = g(x), there are no values of x that satisfy this equation.
- For f(x) > g(x), all x values greater than 4/7 satisfy this inequality.
- For f(x) < g(x), all x values less than 4/7 satisfy this inequality.