To solve these logarithmic expressions in terms of x, y, and z, we can use logarithmic rules and properties.
A. To find the value of log base b of 50 in terms of x, y, and z, we can use the change of base formula. Using base 10 for the logarithm, we have:
log base b of 50 = log base 10 of 50 / log base 10 of b
Since log base b of 2 is x, we can rewrite log base 10 of b as log base 10 of 2 to the power of x:
log base b of 50 = log base 10 of 50 / log base 10 of 2^x
Applying the logarithmic rules, we can express 50 as a product of powers:
log base b of 50 = log base 10 of (2^x * 5^2) / log base 10 of 2^x
Using the properties of logarithms, we can split this expression:
log base b of 50 = (log base 10 of 2^x + log base 10 of 5^2) / log base 10 of 2^x
Since log base b of 3 is y and log base b of 5 is z, we can substitute these values into the equation:
log base b of 50 = (log base 10 of 2^x + 2 * log base 10 of 5) / log base 10 of 2^x
Finally, using the values x, y, and z:
log base b of 50 = (x + 2z) / x
Therefore, the value of log base b of 50 in terms of x, y, and z is (x + 2z) / x.
B. To find the value of log base b of 3000 in terms of x, y, and z, we follow a similar approach:
log base b of 3000 = log base 10 of 3000 / log base 10 of b
Using the change of base formula with base 10:
log base b of 3000 = log base 10 of 3000 / log base 10 of 2^x
Since log base b of 3 is y, we can rewrite log base 10 of b as log base 10 of 3 to the power of y:
log base b of 3000 = log base 10 of 3000 / log base 10 of 2^x * 3^y
Expressing 3000 as a product of powers:
log base b of 3000 = log base 10 of (2^x * 3^y * 5^3) / log base 10 of 2^x * 3^y
Splitting the logarithmic expression:
log base b of 3000 = (log base 10 of 2^x + log base 10 of 3^y + log base 10 of 5^3) / log base 10 of 2^x * 3^y
Substituting the given values:
log base b of 3000 = (x + y + 3z) / (x + y)
Therefore, the value of log base b of 3000 in terms of x, y, and z is (x + y + 3z) / (x + y).