Is this correct?
3ln(2)-1/3ln(x^2+4)
=ln(8/©ø¡îx©÷+4)
I think so but in my notation:
ln [ 8/ (x^2+4)^(1/3) ]
To determine if the expression 3ln(2) - (1/3)ln(x^2 + 4) equals ln(8/(√x + 4)), we can simplify both expressions and compare them.
First, let's simplify the original expression:
3ln(2) - (1/3)ln(x^2 + 4)
Using the properties of logarithms, we know that subtraction between logarithms can be written as division inside a single logarithm.
Hence, we can rewrite the expression as:
ln(2^3) - ln((x^2 + 4)^(1/3))
Now, using the exponent property of logarithms which states that for any real numbers a and b, ln(a^b) = b * ln(a), we can simplify further:
ln(8) - ln((x^2 + 4)^(1/3))
Next, we can use the power property of logarithms to rewrite the expression inside the logarithm as a fraction with a numerator and denominator:
ln(8) - (1/3)ln(x^2 + 4)
Now, let's compare this with ln(8/(√x + 4)):
ln(8/(√x + 4))
We can simplify the expression inside the logarithm, using the property that division is the same as multiplying by the reciprocal:
ln(8 * 1/(√x + 4))
ln(8/√(x + 4))
Both expressions are in different forms, but they are equivalent.
Hence, it is correct to say that:
3ln(2) - (1/3)ln(x^2 + 4) = ln(8/√(x + 4))