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Suppose that a and b are positive numbers for which log, (a) = log15(b) = log25 (a + 2b). What is the value of a​

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Answer: The value of "a" is given by a = 25 - 2b, where "b" is a positive number.

Explanation:

Given that log(a) = log15(b) = log25(a + 2b), we can use the properties of logarithms to solve for the value of a.

Since log(a) = log15(b), we can equate the bases and eliminate the logarithms:

a = 15^log15(b) .....(1)

Similarly, since log(a) = log25(a + 2b), we can equate the bases and eliminate the logarithms:

a = (a + 2b)^log25(a + 2b) .....(2)

Now, we can equate the right-hand sides of equations (1) and (2) since they are both equal to a:

15^log15(b) = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base 15, we get:

log15[15^log15(b)] = log15[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the left-hand side:

log15(b) = log15[(a + 2b)^log25(a + 2b)]

Now, we can equate the bases and eliminate the logarithms:

b = (a + 2b)^log25(a + 2b)

Taking the logarithm of both sides with base (a + 2b), we get:

log(a + 2b)(b) = log(a + 2b)[(a + 2b)^log25(a + 2b)]

Using the property that loga(a^x) = x, we can simplify the right-hand side:

log(a + 2b)(b) = log25(a + 2b)

Since log(a + 2b)(b) = log(a + 2b)/logb(a + 2b) by the change of base formula, we can rewrite the equation as:

log(a + 2b)/logb(a + 2b) = log25(a + 2b)

Now, we can equate the numerators and denominators separately:

log(a + 2b) = log25(a + 2b)

1 = log25(a + 2b)/(log(a + 2b))

Since loga(a) = 1, we can rewrite the equation as:

log25(a + 2b) = log(a + 2b)/(log(a + 2b))

Using the property that loga(a^x) = x, we get:

log25(a + 2b) = 1

This implies that 25^1 = a + 2b, since we are using the definition of logarithm which states that loga(b) = c is equivalent to a^c = b.

Therefore, a + 2b = 25.

Given that a and b are positive numbers, we can deduce that a + 2b > 0.

Solving for a, we get:

a = 25 - 2b

Since a and b are both positive, a = 25 - 2b > 0.

So, the value of a is greater than zero and is given by a = 25 - 2b, where b is a positive number.

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