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If $a^x = b^y = c^z = d^w$, then find the value of $\log_{a} bcd$

According to the data of this logarithm problem, the values of $a$ raised to the power of $x$, $b$ raised to the power of $y$, $c$ raised to the power of $z$ and $d$ raised to the power of $w$ are equal. In this case, the value of the logarithm of product of $b$, $c$ and $d$ to base $a$ is required to find.

Step: 1

The base of the logarithm is $a$. So, try to express the literals $b$, $c$ and $d$ in terms of $a$ purely to replace the literal numbers $b$, $c$ and $d$ in the logarithmic term $\log_{a} bcd$.

Use radical rule and then power rule of exponents to express literals $b$, $c$ and $d$ in terms of $a$.

$(1) \,\,\,\,\,\,$ $a^{\displaystyle x} = b^{\displaystyle y}$ $\implies$ $b = {\Big(a^{\displaystyle x}\Big)}^\dfrac{1}{y}$ $\implies$ $b = {(a)}^\dfrac{x}{y}$

$(2) \,\,\,\,\,\,$ $a^{\displaystyle x} = c^{\displaystyle z}$ $\implies$ $c = {\Big(a^{\displaystyle x}\Big)}^\dfrac{1}{z}$ $\implies$ $c = {(a)}^\dfrac{x}{z}$

$(3) \,\,\,\,\,\,$ $a^{\displaystyle x} = d^{\displaystyle w}$ $\implies$ $d = {\Big(a^{\displaystyle x}\Big)}^\dfrac{1}{w}$ $\implies$ $d = {(a)}^\dfrac{x}{w}$

Step: 2

Now, substitute the values of $b$, $c$ and $d$ in terms of $a$ in the term $\log_{a} bcd$.

$\log_{a} bcd = \log_{a} {(a)}^\dfrac{x}{y} \times {(a)}^\dfrac{x}{z} \times {(a)}^\dfrac{x}{w}$

Use the product rule of exponents to simplify the product of three exponential terms which contain a common base $a$.

$\implies \log_{a} bcd = \log_{a} {(a)}^{\Bigg[ \dfrac{x}{y} \,+\, \dfrac{x}{z} \,+\, \dfrac{x}{w} \Bigg]}$

Step: 3

Now, apply the power rule of logarithms to simplify the log of the exponential term.

$\implies \log_{a} bcd = \Bigg[\dfrac{x}{y} \,+\, \dfrac{x}{z} \,+\, \dfrac{x}{w}\Bigg] \log_{a} a$

Step: 4

The logarithm of number and base are same. So, the logarithm of base is one.

$\implies \log_{a} bcd = \Bigg[\dfrac{x}{y} \,+\, \dfrac{x}{z} \,+\, \dfrac{x}{w}\Bigg] \times 1$

$\implies \log_{a} bcd = \Bigg[\dfrac{x}{y} \,+\, \dfrac{x}{z} \,+\, \dfrac{x}{w}\Bigg]$

$\therefore \,\,\,\,\,\, \log_{a} bcd = x\Bigg[\dfrac{1}{y} \,+\, \dfrac{1}{z} \,+\, \dfrac{1}{w}\Bigg]$

Therefore, it is the required solution for this logarithmic problem.

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