Compare the graphs of the logarithmic functions f(x)= log(7)x and g(x) = log(4)x

Question

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 think x < 0 when f = g and x > 0 when f > g, but when is x when f < g? And how do I explain this?

Please help! Thanks

Answer 1

 

Please visit www.desmos.com and experiment with the function plotter.

You can graph any function of one variable.

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