Solve for x

Problem

(log_16)(x)+(log_4)(x)+(log_2)(x)=7

Solution

1. Apply the change of base formula to express all logarithms in terms of base 2 using the rule (log_a)(b)=(log_c)(b)/(log_c)(a)

(log_16)(x)=(log_2)(x)/(log_2)(16)

(log_4)(x)=(log_2)(x)/(log_2)(4)

2. Evaluate the denominators by identifying the powers of 2, where 16=2 and 4=2

(log_2)(16)=4

(log_2)(4)=2

3. Substitute the values back into the original equation.

(log_2)(x)/4+(log_2)(x)/2+(log_2)(x)=7

4. Find a common denominator of 4 to combine the terms on the left side.

(log_2)(x)/4+(2*(log_2)(x))/4+(4*(log_2)(x))/4=7

(7*(log_2)(x))/4=7

5. Isolate the logarithm by multiplying both sides by 4/7

(log_2)(x)=4

6. Convert to exponential form to solve for x using the definition (log_b)(y)=k⇒bk=y

x=2

x=16

Final Answer

(log_16)(x)+(log_4)(x)+(log_2)(x)=7⇒x=16

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