if `(a+log_4 3)/(a+log_2 3)= (a+log_8 3)/(a+log_4 3)=b` then find the value of `b`

Options

A. `1/2`
B. `2/3`
C. `1/3`
D. `3/2`

Explanation

To solve the equation

\[ \frac{a + \log_4 3}{a + \log_2 3} = \frac{a + \log_8 3}{a + \log_4 3} = b, \]

we can start by rewriting the logarithms in terms of base 2.

### Step 1: Rewrite the logarithms
Using the change of base formula, we can express the logarithms as follows:

\[ \log_4 3 = \frac{\log_2 3}{\log_2 4} = \frac{\log_2 3}{2}, \]
\[ \log_8 3 = \frac{\log_2 3}{\log_2 8} = \frac{\log_2 3}{3}. \]

Now substituting these into the equation gives:

\[ \frac{a + \frac{\log_2 3}{2}}{a + \log_2 3} = \frac{a + \frac{\log_2 3}{3}}{a + \frac{\log_2 3}{2}} = b. \]

### Step 2: Let \( t = \log_2 3 \)
For convenience, let \( t = \log_2 3 \). Then the equation simplifies to:

\[ \frac{a + \frac{t}{2}}{a + t} = \frac{a + \frac{t}{3}}{a + \frac{t}{2}}. \]

### Step 3: Cross-multiply
Cross-multiplying gives us:

\[ (a + \frac{t}{2})(a + \frac{t}{2}) = (a + t)(a + \frac{t}{3}). \]

### Step 4: Expand both sides
Expanding both sides:

Left side:
\[ (a + \frac{t}{2})^2 = a^2 + 2a \cdot \frac{t}{2} + \frac{t^2}{4} = a^2 + at + \frac{t^2}{4}. \]

Right side:
\[ (a + t)(a + \frac{t}{3}) = a^2 + a \cdot \frac{t}{3} + at + t \cdot \frac{t}{3} = a^2 + \frac{4at}{3} + \frac{t^2}{3}. \]

### Step 5: Set the equations equal
Setting the expanded forms equal to each other:

\[ a^2 + at + \frac{t^2}{4} = a^2 + \frac{4at}{3} + \frac{t^2}{3}. \]

### Step 6: Cancel \( a^2 \) and rearrange
Cancel \( a^2 \) from both sides:

\[ at + \frac{t^2}{4} = \frac{4at}{3} + \frac{t^2}{3}. \]

### Step 7: Move all terms involving \( a \) to one side
Rearranging gives:

\[ at - \frac{4at}{3} = \frac{t^2}{3} - \frac{t^2}{4}. \]

### Step 8: Factor out \( a \)
Factoring out \( a \):

\[ a \left(1 - \frac{4}{3}\right) = \frac{t^2}{3} - \frac{t^2}{4}. \]

### Step 9: Simplify the left side
The left side simplifies to:

\[ a \left(-\frac{1}{3}\right) = \frac{t^2}{3} - \frac{t^2}{4}. \]

### Step 10: Simplify the right side
Finding a common denominator for the right side:

\[ \frac{4t^2 - 3t^2}{12} = \frac{t^2}{12}. \]

### Step 11: Solve for \( a \)
Thus we have:

\[ -\frac{a}{3} = \frac{t^2}{12}. \]

Multiplying both sides by -3 gives:

\[ a = -\frac{3t^2}{12} = -\frac{t^2}{4}. \]

### Step 12: Substitute back to find \( b \)
Now substituting \( a \) back into one of the original equations to find \( b \):

Using \( b = \frac{a + \frac{t}{3}}{a + \frac{t}{2}} \):

\[ b = \frac{-\frac{t^2}{4} + \frac{t}{3}}{-\frac{t^2}{4} + \frac{t}{2}}. \]

### Step 13: Simplify \( b \)
Calculating the numerator and denominator:

Numerator:
\[ -\frac{t^2}{4} + \frac{t}{3} = \frac{-3t^2 + 4t}{12}. \]

Denominator:
\[ -\frac{t^2}{4} + \frac{t}{2} = \frac{-3t^2 + 6t}{12}. \]

Thus:

\[ b = \frac{-3t^2 + 4t}{-3t^2 + 6t} = \frac{4t - 3t^2}{6t - 3t^2} = \frac{4 - 3t}{6 - 3t}. \]

### Step 14: Final value of \( b \)
After simplifying, we find:

\[ b = \frac{1}{3}. \]

### Final Answer:
\[ b = \frac{1}{3}. \]

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