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Solve for [tex]\( x \)[/tex].

[tex]\[ 3x = 6x - 2 \][/tex]

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Complete the equation.

[tex]\[ \log _6(6x) = ? \][/tex]


Sagot :

To solve the expression [tex]\(\log_6(6x)\)[/tex], we rely on the properties of logarithms. Here is the step-by-step solution:

1. Identify the expression:
[tex]\[ \log_6(6x) \][/tex]

2. Use the property of logarithms that allows the argument to be split into a product:
The property [tex]\(\log_b(mn) = \log_b(m) + \log_b(n)\)[/tex] states that the logarithm of a product is the sum of the logarithms. Apply this property:
[tex]\[ \log_6(6x) = \log_6(6) + \log_6(x) \][/tex]

3. Simplify [tex]\(\log_6(6)\)[/tex]:
[tex]\(\log_6(6)\)[/tex] is a special case, because any logarithm where the base and the argument are the same is 1. Therefore:
[tex]\[ \log_6(6) = 1 \][/tex]

4. Combine the results:
Substituting back into the expression, we have:
[tex]\[ \log_6(6x) = 1 + \log_6(x) \][/tex]

Therefore, the fully simplified expression is:
[tex]\[ \boxed{1 + \log_6(x)} \][/tex]
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