If (x, y) is on the graph f(x) = log(base10)x, state the coordinates (in terms of x and y) that would be on the graph of f(x) = 2log(base10)(x - 4) + 3?

Can you please explain how to answer this question? Thanks!

remember if the base of the log is 10, we don't have to write the base.

so you have y= logx
and y = 2log(x-4)+3

intersection:
2log(x-4) + 3 = logx
log(x-4)^2 - logx = -3
log ( (x-4)^2 /x ) = -3

(x-4)^2 /x = 10^-3
x^2 - 8x + 16 = (1/1000)x
1000x^2 - 8001x + 16000=0
using the formula
x = (8001 ± 126.5)/2000
= 4.06 or 3.94

but in 2log(x-4) , x-4> 0 , so our 2nd answer is rejected

x = appr 4.06

if x = 4.06, then y = log 4.06 = appr .61

To find the coordinates (x, y) on the graph of f(x) = 2log(base10)(x - 4) + 3, you need to understand the properties of logarithmic functions and transformations.

Here's how you can answer the question step by step:

1. Start with the equation f(x) = 2log(base10)(x - 4) + 3. This equation tells you that the y-values (or output) on the graph are determined by the x-values (or input) with a specific formula.

2. To find the x-coordinate of a point on the graph, you need to solve the equation x = 2log(base10)(x - 4) + 3 for x. This means you need to isolate the x variable.

3. Start by subtracting 3 from both sides of the equation: x - 3 = 2log(base10)(x - 4).

4. Next, divide both sides of the equation by 2: (x - 3)/2 = log(base10)(x - 4).

5. Now, you can rewrite the equation in exponential form. In logarithmic form, log(base10)a = b means that 10^b = a. So, (x - 3)/2 = 10^log(base10)(x - 4).

6. Since the base of the logarithm and the base of the exponential function are the same, they cancel each other out: (x - 3)/2 = x - 4.

7. Multiply both sides of the equation by 2 to get rid of the denominator: x - 3 = 2x - 8.

8. Simplify the equation by subtracting x from both sides: -3 = x - 8.

9. Add 8 to both sides of the equation: 5 = x.

So, the x-coordinate of a point on the graph of f(x) = 2log(base10)(x - 4) + 3 is 5.

10. To find the corresponding y-coordinate, substitute the value of x = 5 into the equation f(x) = 2log(base10)(x - 4) + 3.

f(5) = 2log(base10)(5 - 4) + 3.

Simplify the equation: f(5) = 2log(base10)1 + 3.

Since log(base10)1 = 0, the equation becomes: f(5) = 2(0) + 3.

Evaluate: f(5) = 0 + 3 = 3.

So, the y-coordinate of the point on the graph is 3.

Therefore, the coordinates (x, y) on the graph of f(x) = 2log(base10)(x - 4) + 3 are (5, 3).