Final answer:
The logarithmic equations are solved using properties of logarithms: the logarithm of a quotient, the logarithm of a product, and the logarithm of an exponent. The solutions to the individual parts are x=2.5, an equation to find x, and x=10,000, respectively.
Step-by-step explanation:
We are working with logarithmic equations, which are based on the properties of logarithms. I'll solve each part separately, keeping the properties of logarithms in mind:
(a) ln(x4) − ln(x3) = ln(5x) − ln(2x)
We can apply the property that the logarithm of a quotient is the difference of the logarithms (ln(a/b) = ln a - ln b). Thus:
ln(x4/x3) = ln(5x/2x)
ln(x) = ln(2.5)
So, x = 2.5 after we undo the logarithm.
(b) ln(x − 1) + ln(x + 1) = 2 ln(x + 2)
Here, we use the property that the logarithm of a product is the sum of the logarithms (ln(xy) = ln x + ln y), as well as the property that the logarithm of an exponent is the product of the exponent and the logarithm (ln(xn) = n ln x). Thus:
ln((x − 1)(x + 1)) = ln((x + 2)2)
ln(x2 − 1) = 2 ln(x + 2)
ln(x2 − 1) = ln((x + 2)2)
Now, since the logarithms are equal, the insides must be equal as well:
x2 − 1 = (x + 2)2
This equation can then be solved for x.
(c) log10 x = 4
Using the property that the inverse log or antilogarithm will give us the number itself (10log10 x = x), we get:
x = 104
So, x = 10,000.