Solving the Logarithmic Equation
We are asked to find the value of \(x\) in the equation: \(1 - \log_{10}2 = \log_{10}(5^x + 4^x + 3^x + 2^x + 1)\)
Simplifying the Left Side
First, simplify the term \(1 - \log_{10}2\). We know that \(1 = \log_{10}10\). Using the logarithm property \(\log_b M - \log_b N = \log_b (M/N)\), we get:
\(1 - \log_{10}2 = \log_{10}10 - \log_{10}2 = \log_{10}\left(\frac{10}{2}\right) = \log_{10}5\)Equating Logarithm Arguments
Now, substitute the simplified term back into the original equation:
\(\log_{10}5 = \log_{10}(5^x + 4^x + 3^x + 2^x + 1)\)Since the logarithms have the same base (base 10), their arguments must be equal:
\(5 = 5^x + 4^x + 3^x + 2^x + 1\)Finding the Value of x
We need to find the value of \(x\) that satisfies \(5^x + 4^x + 3^x + 2^x + 1 = 5\). Let's test the value \(x=0\) (Option D):
If \(x = 0\), then:
\(5^0 + 4^0 + 3^0 + 2^0 + 1 = 1 + 1 + 1 + 1 + 1 = 5\)Since substituting \(x=0\) satisfies the equation, \(x=0\) is the correct value.
Final Answer Check
The equation simplifies to \(5 = 5^x + 4^x + 3^x + 2^x + 1\). Testing \(x=0\) gives 1+1+1+1+1 = 5, which is true. Therefore, the value of \(x\) is 0.
Similar Questions
- If \(p + q = 15\), then what is \(q-p\) equal to?
- If \(p + q = 66\), then which one of the following is correct?
For \(x \ge y > 1\), let \(\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right) = k\), then the value of \(k\) can never be equal to
If \(\log_ba = p\), \(\log_dc = 2p\) and \(\log_fe = 3p\), then what is \((ace)^{\frac{1}{p}}\) equal to ?
- What is the number of solutions of \(\log_4(x-1) = \log_2(x - 3)\)?
- Let \(p = \ln(x)\), \(q = \ln(x^3)\) and \(r = \ln(x^5)\), where \(x > 1\). Which of the following statements is/are correct?
I. \(p, q\) and \(r\) are in AP.
II. \(p, q\) and \(r\) can never be in GP.
Select the answer using the code given below. - What is the smallest positive \(x\) satisfying \(\log_{\sin x} \cos x + \log_{\cos x} \sin x = 2\) ?
Important Questions from Logarithms
Given that $132^{0.14} = x$, $132^{0.26} = y$ and $x^z = y^2$, then the value of z is close to:
- If \(p + q = 15\), then what is \(q-p\) equal to?
- If \(p + q = 66\), then which one of the following is correct?
For \(x \ge y > 1\), let \(\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right) = k\), then the value of \(k\) can never be equal to
If \(\log_ba = p\), \(\log_dc = 2p\) and \(\log_fe = 3p\), then what is \((ace)^{\frac{1}{p}}\) equal to ?
