Answer :
Certainly! Let's solve the equation [tex]\(\log_{16} 4 = x\)[/tex] step-by-step.
### Step 1: Understanding the Logarithmic Form
The equation [tex]\(\log_{16} 4 = x\)[/tex] means that we are looking for the exponent [tex]\(x\)[/tex] such that [tex]\(16\)[/tex] raised to the power [tex]\(x\)[/tex] equals [tex]\(4\)[/tex]. In other words,
[tex]\[ 16^x = 4 \][/tex]
### Step 2: Expressing the Base in Terms of Powers
To solve this, it helps to express both 16 and 4 as powers of a common base. Notice:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 4 = 2^2 \][/tex]
### Step 3: Substitute and Simplify
Now substitute these expressions back into the equation:
[tex]\[ (2^4)^x = 2^2 \][/tex]
### Step 4: Simplifying the Powers
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ 2^{4x} = 2^2 \][/tex]
### Step 5: Equating Exponents
Because the bases are the same (both are [tex]\(2\)[/tex]), we can set the exponents equal to each other:
[tex]\[ 4x = 2 \][/tex]
### Step 6: Solving for [tex]\(x\)[/tex]
Now solve for [tex]\(x\)[/tex] by dividing both sides of the equation by 4:
[tex]\[ x = \frac{2}{4} \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[ x = 0.5 \][/tex]
Thus, we have:
[tex]\[ \log_{16} 4 = 0.5 \][/tex]
### Step 1: Understanding the Logarithmic Form
The equation [tex]\(\log_{16} 4 = x\)[/tex] means that we are looking for the exponent [tex]\(x\)[/tex] such that [tex]\(16\)[/tex] raised to the power [tex]\(x\)[/tex] equals [tex]\(4\)[/tex]. In other words,
[tex]\[ 16^x = 4 \][/tex]
### Step 2: Expressing the Base in Terms of Powers
To solve this, it helps to express both 16 and 4 as powers of a common base. Notice:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 4 = 2^2 \][/tex]
### Step 3: Substitute and Simplify
Now substitute these expressions back into the equation:
[tex]\[ (2^4)^x = 2^2 \][/tex]
### Step 4: Simplifying the Powers
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ 2^{4x} = 2^2 \][/tex]
### Step 5: Equating Exponents
Because the bases are the same (both are [tex]\(2\)[/tex]), we can set the exponents equal to each other:
[tex]\[ 4x = 2 \][/tex]
### Step 6: Solving for [tex]\(x\)[/tex]
Now solve for [tex]\(x\)[/tex] by dividing both sides of the equation by 4:
[tex]\[ x = \frac{2}{4} \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[ x = 0.5 \][/tex]
Thus, we have:
[tex]\[ \log_{16} 4 = 0.5 \][/tex]
