Compare the graphs of the logarithmic functions f(x)=㏒_7 x and g(x)= ㏒_4 x. For values of x is f=g, f>g, and f
Please help!
log7(x) = log4(x)
Change of base formula:
logA x = logB x / logB a
Changing to base E:
log7(x) = logE(x)/logE(7) = ln(x)/ ln(7)
log4(x) = lonE(x)/logE(4) = ln(x)/ln(4)
So they are equal when :
ln(x)/ln(7) = ln(x)/ln(4)
cross multplying:
ln(x)*ln(4) = ln(x)*ln(7)
ln(x)*ln(4) - ln(x)*ln(7) = 0 <--- everybody to left side
ln(x) * [ ln(4) - ln(7)] = 0 <--- factors out ln(x)
Let k = [ ln(4) - ln(7) ] is a fixed number constant.
Then,
ln(x) * k = 0
dividing both sides by k, cancels it out.
ln(x) = 0 which means x=1.
So they are equal when x=1
Check:
For x=1
log7(1) = log4(1)
is the same as asking the question:
7^x = 1 = 4^x are both equal to 1 where x is the
same exponent in both....
Of course, any number to power zer0 is 1. So x=1.
The graph in desmos.com agrees with this result.
I graphed y = log7(x) in blue and y = log4(x) in
green.
When x<1, log4(x) < log7(x).
BUT When x>1, log4(x) is larger
Thank you so much!
Oh, logarithmic functions, those sneaky little devils. Don't worry, I'll bring some humor to the math party!
Let's start by imagining we're at a party, and the party is the graph. So, we have our two hosts, f(x) and g(x), ready to show us a good time.
Now, f(x) = ㏒_7 x is like that guy at the party who likes to take things slow. He's not too crazy, and he's always hanging towards the right side of the graph. Meanwhile, g(x) = ㏒_4 x is the adventurous type, always pushing the limits and exploring the left side of the graph.
So, when it comes to values of x where f=g, it's like they've found common ground and decided to party together. Just like when you randomly meet someone at a party and realize you have the same favorite snack. They'll intersect at some point, and that's the sweet spot where f(x) and g(x) collide.
Now, imagine f(x) and g(x) walk up to the DJ booth to request a song. f(x), being the slow and steady type, asks for something calming and low-key. Something like Ed Sheeran's "Perfect." On the other hand, g(x) starts shouting for the DJ to play something fast, like an adrenaline-boosting dance number.
This is where we see f>g. f(x) is always greater than g(x) because it hangs towards the right side and climbs slowly, while g(x) is faster and explores the left side. It's like comparing a tortoise to a hare - slow and steady wins the race!
But wait, where's the other scenario? We need to talk about when f<f. Well, sorry to disappoint you, but that scenario doesn't exist. In this party, f(x) always keeps its head held high and stays above g(x). It's like f(x) has its own VIP section, and g(x) can't quite catch up.
So, to summarize, f(x) and g(x) are two different party animals at the graph party. They intersect at some point (f=g), but f(x) is always greater than g(x) (f>g). As for f<f, well, that's like searching for a unicorn at the party - it's just not gonna happen.
Keep rocking that math party, and remember, there's always fun to be found with graphs and a little bit of humor!
To compare the graphs of the logarithmic functions f(x) = ㏒₇ x and g(x) = ㏒₄ x, let's analyze the behavior of the functions and determine where they are equal, where one is greater than the other, and where one is smaller than the other.
1. Equal values (f=g):
- To find the x-values where f(x) = g(x), we set the two functions equal to each other: ㏒₇ x = ㏒₄ x.
- Using logarithm properties, we can rewrite this equation as: 7ˣ = 4ˣ.
- Taking the logarithm base 10 of both sides, the equation becomes: log₇ (7ˣ) = log₇ (4ˣ).
- Simplifying further, we get: x = 0.
Therefore, both functions are equal when x = 0.
2. f > g:
- To determine where f(x) > g(x), we compare the values of f(x) and g(x) for different x-values.
- Since both functions are logarithms, they approach infinity as x approaches infinity.
- However, f(x) grows at a slower rate than g(x) because 7 > 4.
- Therefore, f(x) > g(x) for all positive values of x.
3. f < g:
- To determine where f(x) < g(x), we again compare the values of f(x) and g(x) for different x-values.
- Both functions tend towards negative infinity as x approaches 0 from the right.
- However, f(x) approaches negative infinity at a faster rate than g(x) because 7 > 4.
- Therefore, f(x) < g(x) for all x-values that are greater than 0 and less than 1.
In summary:
- f(x) = g(x) when x = 0.
- f(x) > g(x) for all positive values of x.
- f(x) < g(x) for all x-values greater than 0 and less than 1.
To compare the graphs of the logarithmic functions f(x) = ㏒₇x and g(x) = ㏒₄x, we need to analyze their behavior and determine when they are equal, when f(x) is greater than g(x), and when g(x) is greater than f(x).
1. When are f(x) and g(x) equal?
To find the x-values where f(x) = g(x), we set the two functions equal to each other:
㏒₇x = ㏒₄x
To solve this equation, we can use the properties of logarithms to rewrite it as an exponential equation:
xⁿ = y ↔︎ n = logₓ(y)
Using this property, we can rewrite the equation as:
7 = 4
Since 7 is not equal to 4, there are no x-values where f(x) = g(x). Therefore, the two graphs do not intersect, meaning their graphs are parallel.
2. When is f(x) greater than g(x)?
To determine when f(x) is greater than g(x), we can compare the values of the two functions for different values of x. We can create a table or use the properties of logarithms to simplify the expressions:
For example, for x = 1:
f(1) = ㏒₇(1) = 0
g(1) = ㏒₄(1) = 0
Since f(1) = g(1), f(x) is not greater than g(x) for x = 1.
For x > 1 (positive values), the logarithm base 7 (f(x) = ㏒₇x) will be greater than the logarithm base 4 (g(x) = ㏒₄x). This is because as x increases, the logarithm base 7 grows faster than the logarithm base 4.
For x = 0, the functions are not defined since the logarithm is undefined for x = 0.
3. When is g(x) greater than f(x)?
Similarly, for x < 1 (positive values), the logarithm base 4 (g(x) = ㏒₄x) will be greater than the logarithm base 7 (f(x) = ㏒₇x). This is because as x decreases, the logarithm base 4 grows faster than the logarithm base 7.
To summarize:
- The graphs of f(x) = ㏒₇x and g(x) = ㏒₄x are parallel and do not intersect.
- For x > 1, f(x) is greater than g(x).
- For x < 1, g(x) is greater than f(x).
- The functions are not defined for x = 0.
To visualize the comparison, you can plot the graphs on a graphing calculator or software to see their respective behaviors.