Question

2.) solve for the value of x.

a.) $4^x = 64$

b.) $log_5 x = 3$

c.) $log_4 1024 = x$

d.) $log_x 3 = \frac{1}{2}$

Explanation:

Part a)

Step1: Express 64 as a power of 4

We know that \( 4^3 = 64 \), so the equation \( 4^x = 64 \) can be rewritten as \( 4^x = 4^3 \).

Step2: Equate the exponents

Since the bases are the same and the exponential function is one - to - one, if \( a^m=a^n \), then \( m = n \). Here \( a = 4 \), \( m=x \) and \( n = 3 \), so \( x=3 \).

Step1: Use the definition of logarithms

The logarithmic equation \( \log_{5}x=3 \) can be converted to its exponential form. By the definition of logarithms, if \( \log_{a}b = c \), then \( a^c=b \). Here \( a = 5 \), \( c = 3 \) and \( b=x \).

Step2: Calculate the value of x

So \( x = 5^3=125 \).

Step1: Use the definition of logarithms

The logarithmic equation \( \log_{4}1024=x \) can be converted to its exponential form. By the definition of logarithms, if \( \log_{a}b = c \), then \( a^c=b \). Here \( a = 4 \), \( c = x \) and \( b = 1024 \). So we have \( 4^x=1024 \).

Step2: Express 1024 as a power of 4

We know that \( 4^5=1024 \) (since \( 4\times4\times4\times4\times4=1024 \)). So \( 4^x = 4^5 \).

Step3: Equate the exponents

Since the bases are the same and the exponential function is one - to - one, if \( a^m=a^n \), then \( m = n \). Here \( a = 4 \), \( m=x \) and \( n = 5 \), so \( x = 5 \).

Answer:

\( x = 3 \)

Part b)