Question

If x, y, z be positive real numbers such that

$\log_{2x} z = 3$, $\log_{5y} z = 6$ and $\log_{xy} z = 2/3$

then the value of z is in the form of m/n in lowest form then find value of n - m

Answer 1

9

Answer 2

The given equations are:

1. $\log_{2x} z = 3$
2. $\log_{5y} z = 6$
3. $\log_{xy} z = 2/3$

We can convert these logarithmic equations into exponential form using the definition $\log_b a = c \iff b^c = a$.

From equation 1: $(2x)^3 = z \implies 8x^3 = z$

From equation 2: $(5y)^6 = z \implies 5^6 y^6 = z$

From equation 3: $(xy)^{2/3} = z$

Alternatively, we can use the change of base formula for logarithms, $\log_b a = \frac{1}{\log_a b}$.

Applying this formula to the given equations:

1. $\frac{1}{\log_z (2x)} = 3 \implies \log_z (2x) = \frac{1}{3}$
2. $\frac{1}{\log_z (5y)} = 6 \implies \log_z (5y) = \frac{1}{6}$
3. $\frac{1}{\log_z (xy)} = \frac{2}{3} \implies \log_z (xy) = \frac{3}{2}$

Now, use the logarithm property $\log_b (MN) = \log_b M + \log_b N$:

From (1): $\log_z 2 + \log_z x = \frac{1}{3}$ (Equation A)

From (2): $\log_z 5 + \log_z y = \frac{1}{6}$ (Equation B)

From (3): $\log_z x + \log_z y = \frac{3}{2}$ (Equation C)

Let $A = \log_z x$ and $B = \log_z y$.

Equation A becomes: $\log_z 2 + A = \frac{1}{3} \implies A = \frac{1}{3} - \log_z 2$

Equation B becomes: $\log_z 5 + B = \frac{1}{6} \implies B = \frac{1}{6} - \log_z 5$

Equation C becomes: $A + B = \frac{3}{2}$

Substitute the expressions for A and B into Equation C:

$(\frac{1}{3} - \log_z 2) + (\frac{1}{6} - \log_z 5) = \frac{3}{2}$

Combine the constant terms and the logarithm terms:

$\frac{1}{3} + \frac{1}{6} - (\log_z 2 + \log_z 5) = \frac{3}{2}$

Find a common denominator for the fractions:

$\frac{2}{6} + \frac{1}{6} - (\log_z 2 + \log_z 5) = \frac{3}{2}$

$\frac{3}{6} - (\log_z 2 + \log_z 5) = \frac{3}{2}$

$\frac{1}{2} - (\log_z 2 + \log_z 5) = \frac{3}{2}$

Apply the logarithm property $\log_b M + \log_b N = \log_b (MN)$:

$\frac{1}{2} - \log_z (2 \times 5) = \frac{3}{2}$

$\frac{1}{2} - \log_z 10 = \frac{3}{2}$

Isolate the logarithm term:

$-\log_z 10 = \frac{3}{2} - \frac{1}{2}$

$-\log_z 10 = \frac{2}{2}$

$-\log_z 10 = 1$

$\log_z 10 = -1$

Convert this logarithmic equation back to exponential form:

$z^{-1} = 10$

$\frac{1}{z} = 10$

$z = \frac{1}{10}$

The value of z is $1/10$. This is in the form m/n, where m = 1 and n = 10.

This fraction is in its lowest form since 1 and 10 have no common factors other than 1.

We need to find the value of n - m.

n - m = 10 - 1 = 9.