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# 7.3 Logarithmic Functions

Which is the inverse of f(x)=3x
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solve the equation
log10((6c+3)/11)=log10(3c/5)
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in astronomy the observed brightness of stars is called apparent magnitude. two stars can be compared using the formula d=2.5logr, where d us the difference of the apparent magnitude of the two stars and r is their brightness ratio. The brightest star Sirius A has an apparent magnitude of -1.46. Alpha Centauri has an apparent magnitude of -2.9.
1) find the difference between the apparent magnitude if Sirius A and Alpha Centauri
2) Find the brightest ratio of the two stars to the nearest tenth

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Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
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f(x) = 3X

f-1(x ) = log 3

log ( 6c +3 ) /11 = log 3c/ 5

If the log of 2 expressions  are equal, then the argument of the log function has to be equal.

6c + 3 / 11 = 3c /5

5( 6c + 3) = 33 c

30c + 15 = 33c

3c = 15

c = 15/3 = 5

### Comments

f(x) = y = 3^x
Swapping x and y: x = 3^y, where y = f-1(x).
There's no way to solve this equation for y = f-1(x) without defining a new function, which we call logarithm. y = f-1(x) = log3(x), which contains all the information as x = 3^y, i.e., the number, x, the base, 3, and the exponent, y.