Solve for k. log1019k=19log10k

Given Equation Simplification: We are given the equation log1019k = 19log10k. We need to find the value of k that satisfies this equation. First, let's use the power rule of logarithms, which states that log_b(m^n) = n*log_b(m), to simplify the right side of the equation.Logarithmic Equivalence: Applying the power rule to the right side of the equation, we get: log1019k = log10(k^19). Now, since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms.Solving for k: Setting the arguments of the logarithms equal to each other, we have: 19k = k^19. Now, we need to solve for k.Checking Solutions: To solve the equation 19k = k^19, we can see that k = 0 is not a solution since 19*0 is not equal to 0^19. We can also see that k = 1 is a solution since 19*1 is equal to 1^19. Let's check if there are any other solutions.Final Solution: Dividing both sides of the equation by k (assuming k is not zero), we get: 19 = k^18. Now, we need to find the value of k that satisfies this equation.The equation 19 = k^18 can only be true if k = 1, because 1 raised to any power is still 1. Any other value of k raised to the 18th power will not equal 19.Therefore, the only solution to the equation 19k = k^19 is k = 1.

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\log_{10}19k = 19\log_{10}k

Given Equation Simplification: We are given the equation $\log_{10}19k = 19\log_{10}k$ . We need to find the value of $k$ that satisfies this equation. $\newline$ First, let's use the power rule of logarithms, which states that $\log_b(m^n) = n\log_b(m)$ , to simplify the right side of the equation.
Logarithmic Equivalence: Applying the power rule to the right side of the equation, we get: $\newline$ $\log_{10}19k = \log_{10}(k^{19})$ . $\newline$ Now, since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms.
Solving for $k$ : Setting the arguments of the logarithms equal to each other, we have: $\newline$ $19k = k^{19}$ . $\newline$ Now, we need to solve for $k$ .
Checking Solutions: To solve the equation $19k = k^{19}$ , we can see that $k = 0$ is not a solution since $19\times 0$ is not equal to $0^{19}$ . We can also see that $k = 1$ is a solution since $19\times 1$ is equal to $1^{19}$ . Let's check if there are any other solutions.
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $\newline$ $19 = k^{18}$ . $\newline$ Now, we need to find the value of $k$ that satisfies this equation.
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $\newline$ $19 = k^{18}$ . $\newline$ Now, we need to find the value of $k$ that satisfies this equation.The equation $19 = k^{18}$ can only be true if $k = 1$ , because $1$ raised to any power is still $1$ . Any other value of $k$ raised to the $18$ th power will not equal $k \neq 0$ $0$ .
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $19 = k^{18}$ . Now, we need to find the value of $k$ that satisfies this equation.The equation $19 = k^{18}$ can only be true if $k = 1$ , because $1$ raised to any power is still $1$ . Any other value of $k$ raised to the $18$ th power will not equal $k \neq 0$ $0$ .Therefore, the only solution to the equation $k \neq 0$ $1$ is $k = 1$ .