Logarithm Problem Explanation

This problem involves solving a system of two logarithmic equations to find the value of an unknown variable, y. We are given:

• Equation 1: logx y = 100
• Equation 2: log2 x = 10

Our goal is to determine the value of y using the information provided. We can solve this by converting the logarithmic equations into their equivalent exponential forms and using substitution.

Logarithm Conversion: Finding x

First, let's use the second equation, log2 x = 10. The definition of a logarithm states that if logb a = c, then it is equivalent to the exponential form bc = a.

Applying this definition to log2 x = 10: The base (b) is 2. The result (c) is 10. The argument (a) is x.

Converting to exponential form, we get: $$x = 2^{10}$$

Logarithm Substitution: Solving for y

Now we know the value of x. Let's substitute this value into the first equation, logx y = 100.

Substitute $x = 2^{10}$: $$log_{2^{10}} y = 100$$

Again, we convert this logarithmic equation into its exponential form. Here: The base (b) is $2^{10}$. The result (c) is 100. The argument (a) is y.

Converting to exponential form, we get: $$y = (2^{10})^{100}$$

Exponent Calculation for y

To find the final value of y, we need to simplify the expression $ (2^{10})^{100} $. We use the rule of exponents which states that $(a^m)^n = a^{m \times n}$.

Applying this rule: $$y = 2^{10 \times 100}$$ $$y = 2^{1000}$$

So, the value of y is $2^{1000}$.